the first reasonably clear definition of continuity may be due to Bolzano, who wrote in 1817 the following definition (in German): f is continuous at x, if and only if "...the difference f(x+h)-f(x) can be made smaller than any given quantity, if h is taken sufficiently small."
In the following proof of the intermediate value theorem he then uses an epsilon for the "given quantity".
Cauchy, writing 6 years later, in 1823, gives the following somewhat less clear definition: " ..the magnitude of the difference f(x+h)-f(x) decreases indeifinitely with that of h."
He clarifies this somewhat as follows: I.e. "an infinitesimal increment in the variable produces an infinitesimal increment in the function itself." where he has explained that an infinitesimal is a quantity whose "successive absolute values decrease indefinitely so as to become less than any given quantity."
the modern definition which needs only to write the letter epsilon for bolzano's "given quantity", as he himself does in his proofs, was published first by heine, a student of weierstrass, in 1874, following lectures of weierstrass.
it is difficult for a modern person to see the real difference in some of these definitions, once one knows what they mean. Indeed I have seen quotations form Newton which sounded essentially like the modern definition of limit. at least one is easily persuaded that the master understood the true meaning very well.
indeed if one spells out the meaning of cauchy's definition as he himself gave it, it is the same. so it seems to me that these workers did themselves understand the meaning of continuity as we do today.
there seems little doubt however that, as has been stated, cauchy did not appreciate the distinction between continuity and uniform continutiy, nor that between convergence and uniform convergence, and made errors or at least omissions of that nature.
the translations i have used here are from "A Source book of classical analysis", Harvard university Press, edited by Garrett Birkhoff.
as to why weierstrass and not bolzano gets credit for epsilon - delta continuity, it seems there is a distinction between originating a concept and influencing others to do so. i.e. bolzano may have led the way himself, but did so in papers devoted almost exclusively to such foundations, so few followed, while others like cauchy were more interested in applying these notions to questions about integrals and series.
thus people interested in cauchy's theorems gave him credit for introducing the new methods he was using, even though those were in fact more primitive than the earlier ones of bolzano. finally it seems weierstrass and his students used almost the same formulation as bolzano but applied it to current topics of interest.
it is odd for example that a theorem could be known as the bolzano - weierstrass theorem, when the two men worked some 50 years apart. possibly it was discovered first by bolzano but rediscovered and popularized by weierstrass. (of course there is also a stone - weierstrass theorem and a riemann - kempf theorem, and a gauss - bonnet - chern theorem,...,in which there is 80-120 years separating the two workers, but in those cases the new results mentioned are significant generalizations of the older ones. In fact a friend of mine once asked Stone just when he had worked with Weierstrass.)
on another thread there is a principle mentioned called there the "arnol'd principle", that if a certain concept or theorem carries a person's name, then it is almost certain that person did not originate that principle. it is then mentioned that indeed arnol'd is not responsible for this principle.