Voltage can still be defined when there are time varying fields. You can research the well known vector potential A. The scalar potential (which is voltage) and the vector potential can be combined into a 4-vector in relativity theory, and a complete field description (including time varying fields) can be provided by scalar and vector potentials. But, this goes a little beyond the thread topic.
The "other" version of KVL could also be called the original version. I say this because Kirchoff's original experiments were with batteries and resistors. The experiments revealed that the sum of EMFs from batteries in a loop equals the sum of the currents times the resistances in the loop. The word potential does not even come up, but it's clear from a modern perspective that resistance times current is a potential drop. Maxwell quotes this version of KVL and gives credit to Kirchoff in his famous Treatise on Electricity and Magnetism. Actually, he mentions both Kirchoff's voltage law and his current law, which nobody argues about at all.
For some reason, this other simplified version of KVL (sum of potentials equals zero) has cropped up in the literature. I'm not sure why, but it is very common to see it in text books. So, it's not too surprising to see Prof. Lewin quoting this as the definition. Again, this is all just semantics, but it is certainly instructive to study and understand the original intent of KVL.
Now it should be said that the original experiments were with EMFs from batteries and not from time changing magnetic flux, but the concept is basically the same. Non-conservative EMFs can be grouped together and used with a very straightforward definition of KVL which says that the sum of EMFs around a loop equals the sum of potential drops around the loop (or some variation on that). Kirchoff and Maxwell define it without reference to potential at all, which can avoid the confusion of what potential means. However, modern theory uses the concept of potential, so it's perhaps better not to avoid it.
To help answer your question, I've attached a copy of Kraus' description of the classical definition of KVL. (Electromagnetics by John D. Kraus, 3rd ed. 1984). Note the footnote at the bottom which mentions time varying fields. As I mentioned above, this is the version of KVL I carry around in my head and use in my professional work. I really don't know why anyone would be interested in the other version of KVL that we commonly see, but who am I to judge?