TL;DR At the time of Ohm's papers (1825-28) current and resistance were very young notions, and Ampere's recent theories shifted their understanding significantly, while Ohm himself completed the shift. Unfortunately, there is no straightforward way to map prior notions to familiar modern concepts. I suppose Kuhn chose this example for a reason, perhaps this is an example of his "incommensurability".
There was no notion of current and resistance to speak of before 1800, the only kinds of current encountered were the electrostatic discharges. Even so, du Fay and Franklin proposed two fluid (positive and negative) and one fluid theories of electricity, which made its flow imaginable. In 1800 Volta invented the first source of continuous current, his famous "pile" of paired copper-zinc plates interspaced with moistened cardboard. Brown's Electric Current in Early Nineteenth-Century French Physics gives a comprehensive survey of the state (or rather evolution) of the art between Volta and Ohm. According to Volta, the electric fluid was propelled into "perpetual impulsion" by an "electromotive contact force" between copper and zinc, multiplied by the pile's layers. However, at the time the only developed theory was electrostatic, and all measurements and measuring instruments were based on it. So Volta's "courant" was an empty word, and his "theory" was soon massaged beyond recognition by an authoritative commission of the French Academy consisting of Laplace, Coulomb, Fourcroy, Guyton, and Biot.
The report reflecting the commission's views was written by Biot, and unsurprisingly described the pile not as a current source but as a sequencer of electrostatic discharges. But with it, unlike with Volta's theory, one could do measurements of the "quantity of electricity" (charge) and the "exciting force"/"intensity"/"tension" (roughly voltage between the terminals of the pile) with electrostatic instruments, like Coulomb's electroscope. To the extent that resistance figured there Biot identified it with "isolating lengths" in the pile. Biot's electric quasi-statics remained the orthodoxy until 1820s, and within it there was no way to even measure the current with any precision. The winds of change came with Oersted's discovery of the current affecting magnetic needles, and the invention of the galvanometer based on it. But the change maker was Ampere. As an outsider to the French scientific establishment he was less reserved about dismissing Biot's quai-static picture, and going back to Volta's original intuitions, but with better instruments.
Ampere introduced the idea of a perpetual circular movement of the two electricities in the wire somehow started by the pile, "reunions in the wire" as he called them. It is interesting that he first came up with his famous hypothesis of molecular currents in a magnet, the reunions in infinitesimal circles around each molecule, where it was obvious that the motion has to be continuous to prevent the immediate recombination, and only then transferred the idea to voltaic circuits. Ampere's theory was a hybrid though, as it sharply separated electrostatic effects in the pile, and electrodynamic ones in the wire. The tension of the pile was seen as of electrostatic nature, associated with attraction/repulsion and sparks, and when the pile's terminals were connected by a wire it was seen (and measured) to vanish. The current was seen to produce chemical and magnetic effects. A well known experiment, where several piles are placed together in a series, produced magnified tension, but essentially the same current as measured by the magnetic needle. Never the twain shall meet.
It was this Ampere's distinction that Ohm disturbed in 1827 with his law. To his fellow physicists he appeared to be suggesting something like a formula linking the intensity of light to its frequency, two clearly independent variables. Here is from Schagrin's Resistance to Ohm's Law:
"...in having the magnetic action of the electric stream depend on the exciting force of the source, Ohm appeared to be confounding the well recognized distinction between tension electricity and current electricity... Ohm of course was able to explain away the observations that led to the distinction between tension and current electricity. But a scientist who had worked with this distinction, who had obtained fruitful results on the basis of it, could scarcely be expected to shift his viewpoint so radically. To this scientist Ohm's experimental results could not confirm the law; Ohm's remarks were not so much false as they were irrelevant (or even worse, a misuse of language)."
In a sense, one has to presuppose something like the Ohm's law to get the modern definitions of current, and especially voltage and resistance, their modern meanings in part depend on its form.
"Ohm initiated another, more serious, conceptual innovation, of which the first signs can be found in these earlier papers. This was a further revision of the notion of Spannung such that each point in a closed circuit
had a particular Spannung [tension]. The Spannung of a source was, in these terms, the difference in Spannung between the terminals - a notion similar to the concept in the older theory. But that each point of a closed circuit had a Spannung was a notion entirely foreign to the older theory. With this revision of the meaning of "Spannung" Ohm was able to use his law on segments of a circuit rather than on the entire circuit at once."