What is the olden name for sideburns?

Upon search, I found out that sideburns has evolved from burnsides, named after the Civil War veteran and Rhode Island senator Ambrose Burnside. See here

But surely sideburns existed before him, so what where they called then? Simply beard on cheeks/sides of face?

The style shown in your link was called...

mutton-chop beard

also

mutton-chop whiskers

Mutton chop beards are so named because they resemble a piece of chopped mutton, particular when shaped correctly. They’re characterized by sideburns that are chopped, or cut, along with the lower jawline and extend to the chin.

This only refers to the luxuriant type of facial hair style as pictured in your link. It is still in use today.

The phrase seems to go back to approximately 1845, according to N grams.

However, some probably just referred to them as

whiskers

Notice that in the descriptions in Prints of English Heads, "whiskers" is listed separately from styles of beards.

…Black Cap, Hair, Whiskers, peaked Beard, Band…

In the next transcript, from the The Conspiracy Trial for the Murder of the President: And the ..., Volume 2

Q: Do you think he has a mustache only, or a mustache and side-whiskers?

A: I cannot undertake to say that were any side-whiskers...

So it appears that what we call side-burns now was originally called either side-whiskers,or mutton-chop whiskers (also mutton-chop beard) , depending on the style. What I found confusing in the search was that "whiskers" was also used to describe a mustache.

Beards have come and gone in a variety of styles for men, but the "sideburn" was probably only seen after 1800, and then only as part of a beard.

The OED confirms BURNSIDE as a precursor to sideburns.

‘A style of beard such as that affected by General Burnside (1824–81), consisting of a mustache, whiskers, and a clean-shaven chin’ ( Cent. Dict. Suppl. 1909). Frequently plural. Also attributive. Cf. sideburn n. 1875 Cincinnati Enquirer 6 July 2/1 His whisker was of the Burnside type, consisting of mustache and ‘muttonchop’, the chin being perfectly clean.

1881 I. M. Rittenhouse Jrnl. in Maud (1939) i. 36

The older one has lovely burn-sides.

1907 Outing 50 279 Such

various patterns of ornamental whiskers as the ‘Piccadilly Weeper’ (No. 2), the ‘Burnside’, etc.

1930 Publishers' Weekly 8 Feb. 679

In the days of copper-toed boots and burnsides..our grandfathers were buying this book.

Sideburns Etymology:

Apparently an alteration of Burnside n., after side-whisker n. at side n.1 Compounds 3, side hair n. at side n.1 Compounds 1c(b), etc.

orig. U.S. Thesaurus » Categories »

A strip of facial hair grown by a man down each side of his face in front of the ears. Usually in plural. 1876 People (Indianapolis) 8 Apr. 2/4 Norris and Warner want to be fashionable. They are cultivating side-burns.

1887 Chicago Jrnl. 1 Aug. McGarigle has

his mustache and small sideburns still on.

1936 G. Greene Journey

without Maps ii. iv. 197 He was..handsome in his native robe and his sideburns.

1985 Times 31 Jan. 13/5 The world will not be won or

lost on the fall of a fringe or the length of a sideburn. 2004 T. C. Boyle Inner Circle ii. iv. 298 The manager was a very proper-looking character with swept-back hair, silvered sideburns and the trace of an Italian accent.

The text in bold above - re "side-whiskers" and "side-hair" answer the question.

If I'm not mistaken, long side-whiskers were also known as "dundrearies" (the term seems to be early 20th century at latest).