Definition of "kissing cousins"— Are the dictionaries wrong/incomplete?

With relatives in the US south, I always thought that the definition of "kissing cousin" was a second cousin (or more distant) whom you could kiss and subsequently marry (FWIW I never did either!).

However, a number of dictionaries have a very different definition: namely, a relation close enough to kiss on meeting (sort of like a hug, I gather). I never heard the term used this way. Is it a recent "invention"?

Example:

kissing cousin
noun
1. A relative close enough to be kissed in salutation, hence anyone with whom a person is fairly intimate:
The two species will often prove to be kissing cousins, for they'll crossbreed.
You guys talk like kissing cousins

TFD and Oxford Dictionaries confirm The Dictionary of American Slang's definition.

The closest reference I found to the idea I mentioned was the discussing of Cousin Marriage in Wikepedia.

Has any else heard the term used to refer to cousins who can be married?

The term usually means a blood relation who is distant enough that you can fool around with, or indeed even marry / have children with.

Your actual question:

Are the dictionaries wrong/incomplete?

Yes, this is an unusual case where, apparently, all the reference works are just plain wrong.

Yes, the reference works mentioned are completely, totally, wrong.

(Note: the idea that the phrase related to "greeting procedures" is totally nonsensical. The idea of Americans (now or historically) "kissing" in greeting is absurd.)

So, say a child "played doctor" with a full sibling, or a full first cousin. That would be incredibly disturbed and psychologists would be called-in.

But a "kissin' cousin" is a relative - distant enough - where it's NOT a psychological emergency if there is some mild sexual involvement.

The term cheekily suggests the frisson of (very mild) incestuous sexuality.

Like any term, of course, it is used in different ways:

(*) distantly related enough that kids can "play doctor"

(*) distantly related enough that two people can indeed have full unprotected sexual intercourse

(*) distantly related enough that, legally, two people can get married

You need only look at the mentioned Elvis song which has astonishingly sexually raw lyrics.

It - uh - playfully talks about light incest, for an example of the usage of the phrase in question:

https://www.youtube.com/watch?v=hn0EdIy_OhI

Well I've got a gal, she's as cute as she can be
She's a distant cousin.
But she's not too distant with me
We'll kiss all night
I'll squeeze her tight
But we're kissin' cousins
and that's what makes it all right
All right, all right, all right

My God, lyrics were explicit then.

'Kissing cousins' in reference works

Reference works vary considerably in how broadly or narrowly they understand the term kissing cousins. On the one hand we have this entry from Christine Ammer, The American Heritage Dictionary of Idioms (1997):

kissing cousins Two or more things that are closely akin or very similar. For example, They may be made by different manufacturers, but these two cars are kissing cousins. This metaphorical term alludes to a distant relative who is well known enough to be greeted with a kiss. {c. 1930}

Here, although she acknowledges the figurative use of "kissing cousins," Ammer sees the origin of the term as being strictly the well-known distant relative. In contrast, Harold Wentworth & Stuart Flexner, Dictionary of American Slang (1960) has a long, fairly elaborate entry for "kissing cousin":

kissing cousin 1 A constant companion or friend, of the same or of the opposite sex, who is granted the same intimacy accorded blood relations. 1951: {same sex} "You guys talk like kissing cousins." Movie, The Tanks Are Coming. --> 2 Specif., a close platonic friend of the opposite sex. --> 3 Humorously, a member of the opposite sex with whom one is sexually familiar when the parties believe their intimacy is unknown. Orig. the term implied blood relationship and still does when used in Southern hill dial. In the South during the Civil War, kissing cousins were relatives who had the same political views. 4 A facsimile, someone or something closely resembling someone or something else.

Definition 4 in Wentworth & Flexner is evidently the same as the primary definition in Ammer. The completely rewritten (by Robert Chapman & Barbara Kipfer) Dictionary of American Slang, third edition (1995) hews much closer to Ammer than to Wentworth & Flexner on this question:

kissing cousin (or kin) by 1940s 1 n A relative close enough to be kissed in salutation, hence anyone with whom a person is fairly intimate: [example omitted] 2 n A close copy: [example omitted]

'Kissing cousins' in Google Books search results

That Ammer is correct as to the original meaning (though wrong as to the date of origin) of the phrase is clear from early Google Books matches for "kissing cousin." From Edward Pollard in a letter from Oakridge Virginia (1858), in Black Diamonds Gathered in the Darkey Homes of the South (1859):

Here I spent a few days of delightful happiness, especially in company with my pretty cousin with the Roman name. But having found out that kissing cousins was no longer fashionable in Virginia, and that it excited my dear aunt's nerves, with one last lingering kiss of the sweet lips, I had my little leather Chinese trunk packed on the head of a diminutive darkey and again embarked upon the James river and Kanawha canal.

From Edward Pollard (again), "A Re-Gathering of 'Black Diamonds' in the Old Dominion," in Southern Literary Messenger (October 1859):

Pursuing my journey, I make the usual round of visits to uncles and cousins, and even remoter relatives. Again I am charmed by visits to hospitable kin; and again, I am especially charmed by the Virginia fashion of kissing cousins to the third degree. The pretty cousin “with the Roman name” is again greeted with a kiss, and found not only on her lips but in her heart as sweet as ever. God bless her!

From Julian Street, American Adventures: A Second Trip "Abroad at Home" (1917):

Speaking broadly of the South, I believe that there survives little real bitterness over the Civil War and the destructive and grotesquely named period of "reconstruction." When a southern belle of to-day damns Yankees, she means by it, I judge, about as much, and about as little, as she does by the kisses she gives young men who bear to her the felicitous southern relationship of "kissing cousins."

And from WPA Writers' Program, Maryland: A Guide to the Old Line State (1940):

Marylanders who can trace their ancestry to the early period of colonization are all cousins, the outsider quickly concludes. Most of them actually are 'connections,' and when they aren't, they are 'kissing cousins,' which generally means that parents and grandparents were lifelong, intimate friends.

In none of this usage is there a hint of "kissing cousins" being used to refer to relatives who can kiss without taboo because they are distantly enough related that marriage is legally available to them. In fact, kissing has never been taboo between close relatives. The earliest Google Books instance I can find that connects "kissing cousins" with marriage is a 1967/1968 issue of Health News [combined snippets]:

Kissing Cousins

Is it against the law in New York State for first cousins to marry? If they do, will there be something wrong with their children?

New York State law does not forbid marriage between first cousins. There is a somewhat higher risk that children resulting from such a marriage may be born with a genetically determined defect or disease than would be present in children resulting from a marriage between two individuals who are not related.

But this instance involves an eye-catching subhead, not an attempt to define kissing cousins in terms of a level of consanguinity at which marriage is acceptable. The earliest actual usage I could find of "kissing cousins" in the sense of "blood relatives who are eligible to marry one another" is in Richard Jensen, Illinois: A History (1978):

The churches enhanced their cohesiveness by fostering marriages within the group. This actually widened the range of eligible spouses from immediate neighbors and "kissing cousins" to unrelated persons. The frontierspeople intermarried freely with natives of other states (except Yankees and foreigners, who rarely gave or took brides from their upland southern neighbors in Illinois).

'Kissing cousins' in newspaper database search results

The Library of Congress's Chronicling America database of old newspapers finds a few matches for "kissing cousins" for the period between 1834 and 1922, the most interesting of which is "Kissing a Pretty Cousin," in the [Montpelier] Vermont Watchman and State Journal (August 28, 1845):

It is a grave question—has a man a right to kiss the tempting lips of a pretty cousin? The Philosopher of the Richmond Star averreth that he has, on being provoked to do so—and they say, he is the easiest man to be provoked within the limits of the "Old Dominion." But he says that the lips of a pretty cousin are a sort of neutral ground, between a sister's and a stranger's. If you sip, it is not because you love, not exactly because you have the right, not upon grounds Platonic, nor with the calm satisfaction that you kiss a favorite sister. It is a sort of hocus-pocus commingling of all, into which each feeling throws its parts, until the concatenation is thrilling, peculiar, exciting, delicious, and "emphatically sleek." This is as near to a philosophical analyzation as he can well come, he thinks, and then he intimates that all the sweet, pretty girls are kissing cousins in Virginia. The Major says he hopes this custom will travel fast into the other States, and become extensively fashionable—and the Major is a man of taste.

A shortened version of the original remarks of the Virginia authority—identified as "Corporal Streeter"—appears in The [Spartanburg, South Carolina] Spartan (September 25, 1844).

Conclusions

Perhaps the most interesting thing about the OP's question is how widespread the notion is that "kissing cousins" has the meaning "cousins distantly enough related to be eligible to marry each other," despite the absence of support for that meaning in reference works. In effect, we have a regional (Southern) American meaning—"related closely enough to justify kissing at greeting"—of long standing (going back at least as far as 1844 in Virginia) that subsequently caught on elsewhere as a phrase with a completely different meaning, without the newer users' having a clear notion of what the phrase originally meant.

Under the circumstances, it's hard to say how well established the "marriageable" sense of "kissing cousins" is. That meaning, though unconfirmed by reference works, shows signs of being fairly widespread today—as we see from the fact that the poster and several answerers here (including at least one from the U.S. South) seem to share it.