Alright, IIRC R1 = left, R2 = right, A1 is top of R1.......D2 is bottom of R2. So if we are measuring R2 with a voltmeter on the right, we are also measuring the equal voltage comprised of the contribution of the loop and R1. Assuming clockwise current flow, the voltage calculated across R2 is I*R2 and the other matching voltage would be the EMF (A2-A1) - (I*R1) - EMF (D1-D2). Or, if I've botched the signs, something like that. Is that right?
I have no idea what the points A2, A1, D2, D1 refer to - probably they are present in the original Lewin drawing but I do not have it at had now. But I am a bit troubled by your use of "EMF (A2-A1)" and I believe therein lies the rub. You are still trying to apply Kirchhoff and you are implicitly assuming the wires are like 'batteries'. No, you need to let go of that because there is no longer the Eind field in the wires. It has been obliterated by the Ecoul field. This is what KVLers have trouble accepting. For example, thinkfat in the message above is trying to avoid bringing the charges into the discussion, but it's all about charges. Literally, all. And I mean "ALL".
You can try to avoid mentioning them directly but current and voltage are derived quantities to the more fundamental physical quantities charge and flux. Sooner or later you have to deal with the complete picture.
What you need to do is to apply the correct laws: KVL in the outer loop (because it is free of varying magnetic flux) and Faraday in the ring (because it links magnetic flux). When you apply Faraday in a circuit you apply the integral form and you put in the EMF linked by the closed (measurement) loop. All of it, because it is the surface integral of the the area whose boundary is your measurement loop.
You can see it how I solved the circuit in my second (or third, I don't remember) post about this topic on EESE linked above.
You can also see it applied by prof. Sam Ben-Yaakov of Ben Gurion University
Intuitive analysis of non conservative electrical circuits and an answer to a Riddle As you can see, Ben-Yaakov seems to question Lewin's statement that KVL is for the birds, but only because he acknoweldge that you can still use KVL in the outer loops. But in the ring, and in the measurement loops that run around the core, he uses Faraday.
So KVL is no longer generally applicable. It really is for the birds, when you consider the ring alone.
So how does this work when you have only a partial turn or a wire that goes straight through a core like a current transformer?
A partial turn of wire is not a closed circuit, so it cannot have a current flowing. What you have is that the Eind field will displace charge, but charge has its own coloumbian field and it will try to resist this displacement. In the end, instant after instant, you reach (with relaxation times) an equilibrium where the induced field Eind in the conductor is exactly countebalanced by the coloumbian field Ecoul generated by the charge that has accumulated at the terminals (and a bit on the lateral surface, depending on geometry).
Things break down when the field change so fast that you cannot reach an equilibrium over the whole extension of the system. But this falls out of the boundaries of magneto-quasistatics. I don't want to go retarded on you.
Again, you cannot do away with the role of charges. And Eind alone tells only half of the story.
Your follow-up question - because I've developed this ability to read minds - is "what about partial inductances? there are books on partial inductance written by prominent scholars in the field of EMC, what about them?"
It is a very useful concept but needs extreme care not to be misunderstood. I consider it a useful bean-counting tool to see where most of the Eind field is captured by a particular loop, but it certainly does not authorize you to think there is a voltage developed along (note that I am using 'along') any part of a coil or conductor in a loop. I had a fourth answer for EESE in the making that will address this topic briefly but I no longer actively participate in the forum. I might make an exception because this madness about KVL is really irking, and the more explanation are out there, the better for all.