As far as I can tell, the idea of “same size means establishing a bijection” appears (in a formal sense) at essentially the same time as it was discovered that

As far as I can tell, the idea of “same size means establishing a bijection” appears (in a formal sense) at essentially the same time as it was discovered that there are pairs of infinite collections where you can’t do that (i.e. with Dedekind and Cantor in the late 1800s), so I don’t think that second definition ever actually existed in a meaningful sense, although I think your intuition seems reasonable (albeit “bending” holds the wrong connotation, imo, as this would seem to be a rather principled generalization of an existing concept rather than an arbitrary decision among a number of somewhat reasonable options). Regardless, I’d be very interested to read a survey of the history of ideas about cardinality/ordinality if you have a good one, since modern treatments of these ideas are radically different than how humans understood them even 150 years ago.