Is the use of "conversely" to mean "on the other hand" correct?

Reference-book definitions of 'conversely'

Merriam-Webster's Eleventh Collegiate Dictionary (2003) defines the noun converse and the adjective converse as follows:

converse n (1570): something reversed in order, relation, or action: as a : a theorem formed by interchanging the hypothesis and conclusion b : a proposition obtained by interchanging of the subject and predicate of a given proposition {"no P is S" is the converse of "no S is P"}

converse adj (1794) 1 : reversed in order, relation, or action 2 : being a logical or mathematical converse {the converse theorem}

The definitions seem reasonable enough, but the extent to which they accurately reflect how people actually use the noun and adjective converse and (more especially) the adverb conversely depends on how broadly we interpret the meaning of the phrase "reversed in order, relation, or action."

Bryan Garner, Garner's Modern American Usage, third edition (2009) outlines the traditional logical senses of various members of the -verse family of words quite clearly:

converse, n.; obverse, n.; inverse, n.; reverse, n. In logic, these words denote various types of opposition. Converse = a statement derived from another statement by transposing the terms on each side of an antithesis {honor without courage; courage without honor} Obverse = an equivalent statement made by negative inference {no person is immortal; all people are mortal}. Inverse = a statement made by inference from an original negative proposition by changing the subject but keeping the predicate the same {no comedy is drama; some noncomedies are drama}. Reverse, the broadest of these terms, means simply "the contrary" and embraces the other three.

But in everyday situations—as opposed to in discussions of mathematical relationships or in analyses of logical forms—English speakers rarely say obversely when they mean "as an equivalent statement made by negative inference," or reversely when they mean "to the contrary." Instead, the most common way to say these things in a single word is to use contrarily or conversely.

Real-world, nonspecialist usage of 'conversely' in the 1800s (and before)

An Ngram chart for the years 1800–2000 strongly suggests that the frequency of occurrence of conversely (blue line), obversely (red line), and reversely (green line) doesn't reflect the relative commonness in human thought of the underlying logical relationships "proposition with transposed terms," "proposition based on negation of a prior proposition's terms," and "proposition expressed generally as the reverse of a prior proposition":

In geometry, converse relationships are undoubtedly of much greater interest than obverse relationships; and reverse relationships (which, as Garner says, encompass both converse and inverse relationships) form too broad a designation to be especially useful in mathematical contexts. So conversely (along with inversely and directly) was much more common than obversely or reversely in the nineteenth century, a period when (to judge from Google Books search results) the terms appeared primarily in the context of mathematics or formal logic. A typical example, from Thomas Duncan, Elements of Plane Geometry (1848):

COR[OLLARY] 1. Given the diameter or radius, we can find the circumference very nearly; and conversely.

But when people began looking to the -versely family of words for a way to describe everyday, nonmathematical relationships involving a reversal of terms, or of actions, or of viewpoint, conversely appears to have appealed to them more than inversely did. One striking (and quite early) example of this preference is in David Hartley, Observations on Man, His Frame, His Duty, and His Expectations (1749):

There is not an atom perhaps in the whole universe, which does not abound with millions of worlds ; and, conversely, this great system of the sun, planets, and fixed stars, may be no more than a single constituent particle of some body of an immense relative magnitude, &c. In like manner, there is not a moment of time so small, but it may include millions of ages in the estimation of some beings ; and, conversely, the largest cycle which human arc is able to invent, may be no more than the twinkling of an eye in that of others, &c.

Here, it seems clear that Hartley is using conversely to mean "looked at from the opposite extreme or point of view"—which is essentially the meaning that the Free Dictionary (cited in the OP's question) reports with its definition "In a contrary or opposite way; on the other hand."

Alexander Bain, English Composition and Rhetoric: A Manual (1867) criticizes the use of conversely to indicate negation:

Besides the regular conjunctions, there are a variety of words and phrases serving for reference.

Thus the expressions for the very important ends of stating opposition or negation, involve a reference to what went before ; —On the contrary, on the other hand, conversely, obversely. Of these, the only one properly signifying negation is the first ; the others are frequently misused for that signification. "On the other hand" properly implies an alternative. "Conversely" is, in strict logic, transposing the terms of a proposition (Some Englishmen are wise ; some wise men are Englishmen). "Obversely" denies the opposite of a proposition (All men are mortal, no men are immortal), which is to re-affirm it from the other side.

Modern real-world, nonspecialist usage of 'conversely'

But a century later, it was not at all uncommon to see conversely used to indicate negation (in the sense of "to the contrary") or an alternative (in the sense of "on the other hand"), or simply to provide a transition word of no clear meaning. Here are recent examples of each.

From Paul Ellefson, Calder Hibbard, and Michael Kilgore, Federal and State Agencies and Programs Focused on Nonfederal Forests in the United States: An Assessment of Intergovernmental Roles and Responsibilities (2003) [combined snippets], where conversely seems to mean "nevertheless" or "to the contrary":

Most often mentioned as of little or no importance as a deterrent to accomplishment of a desired outcome was apathy toward program innovation, unhealthy competition between state and federal governments, lack of clearly defined state and federal roles involving nonfederal forests, and excessive state reliance on federal assistance. Although these factors lead in the frequency with which they were identified as of little or no importance, they were (conversely) considered very or somewhat important by a sizeable portion of the respondents (Table 30).

From Robert Mainfort, Indian Social Dynamics in the Period of European Contact: Fletcher Site Cemetery, Bay County, Michigan (1979) [combined snippets], where Conversely appears to be standing in for "In contrast" or "On the other hand":

In an egalitarian society, the death of a child, who has not had an opportunity to develop numerous and significant social identities, will affect primarily its close relatives and, hence, exhibit few components in its significata. Conversely, the death of a group leader would involve identity relationships not only with more individuals, but also more groups, would result in the manifestation of more components in his significata.

And from Charles Herdendorf, C. Nicholas Raphael, and Eugene Jaworski, The Ecology of Lake St. Clair Wetlands: A Community Profile (1986):

Although methodologies are mixed, Table 2 is useful in establishing economic values within an order of magnitude and in priority. Notice that nutrient control, sport fishing, and fish production are clearly of more value than the traditional uses, i.e., waterfowl hunting, trapping, water supply, and commercial fishing. Conversely, these data are presented as average economic values, even though not all wetlands are alike. Moreover, the methodologies did not consider surplus value, as would be assessed by the willingness to pay technique.

It's hard to say what Conversely is doing in the middle of this long paragraph, other than perhaps signifying "Moving on to another point..."

Conclusions

In everyday English, conversely has a wide range of meanings—including "on the other hand," "in contrast," "nevertheless," "from a very different point of view," and "to the contrary"—in addition to its narrow traditional meaning of "with the terms transposed," in mathematics and formal logic.

The reluctance of some dictionaries to admit the legitimacy of such everyday usage, despite evidence that the usage has persisted over many decades, is baffling. In any case, the real-world usage shows no sign of abating, with or without dictionary approval.