Aside from infinitesimals, another "metaphysical difficulty" of the time were Newton's fluxions. For instance, Lagrange (Fonc. Anal., 1797, pp. 3–4) comparing fluxions to infinitesimals ("differential calclulus"): "This method, whose calculus agrees in fundamentals and operations with the differential calculus, differs only in the metaphysics that seems in effect clearer, because everyone has, or believes to have, an idea of velocity." [I read "seems...clearer" as casting doubt on its clarity.]
As for defining infinitesimals in terms of limits, here is Cauchy in Cours d'analyse (1821):
Lorsque les valeurs successivement attribuées à une même variable s'approchent indéfiniment d'une valeur fixe, de manière à finir par en différer aussi peu que l'on voudra, cette derniére est appelée la limite de toutes les autres....Lorsque les valeurs numériques successivement attribuées à une même variable décroissent indéfiniment, de manière à s'abaisser au-dessous de tout nombre donné, cette variable devient ce qu'on nomme un infiniment petit ou une quantité infiniment petite. Une variable de cette espèce a zéro pour limite. --- Cauchy, 1821, p. 4
On dit qu'une quantité variable devient infiniment petite, lorsque sa valuer numérique décroit indéfiniment de manière à converger vers la limite zéro. --- Cauchy, 1821, p. 26
The first quote is from the "Preliminaries," which is a review or overview of certain types of quantities and other things, such as limits and infinitely small quantities mentioned in the passage. The second quote is from chapter 2 on infinitely small and infinitely large quantities. Cauchy restates the definition of an infinitely small quantity.
In the first passage, limits are defined first, and the definition of an infinitely small quantity is what we called a variable "when the numerical values successively attributed to the same variable decrease indefinitely, so as to fall below any given number." Cauchy notes this type of number has zero as a limit, which is clear because the definition of a limit is the same except (1) it is the difference between the quantity and a fixed value (limit) that decreases and (2) the decrease in the difference is characterized as "to end by differing as little as one would wish."
In the second passage, Cauchy defines an infinitely small quantity in terms of limit: "A variable quantity becomes infinitely small when its numerical value decreases indefinitely so as to converge to the limit zero."
There is so little difference conceptually in the characterizations of limit and infinitesimal, it is not clear that it is important to determine which is defined in terms of the other, or, as it seems to me, both are defined in terms of evanescent, successive values. This last idea is an important idea in Cauchy's calculus. Since that idea is coextensive with an "infinitely small quantity", it makes some logical sense to define limit in terms of infinitely small, which Cauchy could have done. But he did not, at least not in Cours d'analyse.
As for purpose, Cauchy writes in his introduction, which he later characterizes as a search "to perfect mathematical analysis":
As for the methods, I have sought to give them all the rigor required in geometry, so as never to resort to reasons drawn from the generality of algebra. --- Cauchy, 1821, p.ii
About the generality of algebra, he says:
The reasons for this type can only be considered, it seems to me, as inductions capable of presenting the truth sometimes, but which are hardly consistent with the exactitude so vaunted of the mathematical sciences. --- Cauchy, 1821, p.iii
This probably is also one of the "metaphysical difficulties" Wallace refers to.
