You can not measure two different voltages when you measure in the exact same two points at the exact same point in time.

Aaaaand... we're back to square one.

Please point out what part of my short statement is wrong.

And if that is not wrong then what was the point of that experiment?

I do have to agree with Sredini this time.

The problem is that this ideal voltmeter still needs wires to connect to the points of interest on the circle and that's how you can get different voltages depending on what path these wires take, they are as much part of the circuit as the loop itself.

If you take the formal definition of voltage that says that the voltage is the integral of all forces acting on electrons along a path. Then you do get a different number depending on what path do you take trough the loop. By definition there are indeed two possible voltages across the two points. This is because both the electric field of charge separation and the magnetic EMF count towards the total voltage.

Voltmeters only read the charge separation part of the voltage because that's what drives current trough its internal resistance. If you connect the voltmeter with the wires taking such a path that the magnetically induced EMF is zero you get to measure that charge separation and you get the single result you expect (The result is indeed 0.4V). This is the same result that circuit mesh analysis will give (that is called KVL here, but its more than just KVL).

The voltage from charge separation is always defined as a single number for all points in any circuit. Its simply how many electrons are sitting there. More electrons more negative the voltage. Its only the magnetic EMF part of the voltage that depends on what path you take, this is because that EMF is pulling the electrons into a certain direction. This makes electrons motivated to move in that direction even if there is already the same amount of electrons there, but only in that direction.

So to conclude yes there are two voltages across A and B in Dr. Lewins circuit according to the formal definition of voltage, but this is not useful voltage that can be harnessed, its just a incomplete calculation of voltage that requires the rest of the loop to be added in too and that then gives you a single result. In your case that is a single result of voltage across the voltmeter terminals.

These multiple voltage across two points in electrical engineering do sort of the same thing as complex numbers in math. The imaginary part of complex numbers don't really physically exist, but it is there to make the math work out that otherwise wouldn't be possible(Such as square roots of negative numbers). Same goes here with Dr. Lewins example. The voltages are indeed there according to the math, but its not a real voltage you can "physically touch" in the real world. Much like a voltage of 6 V or 5.66+j2 V look the same to a voltmeter but are not the same in the math.

EDIT: Note there are no complex numbers involved in Dr. Lewins example. I was just making a comparison to math. Tho if you want you can still stick complex values of voltage and current into Kirchhoffs circuit laws and it still works(very useful when you have AC sources and reactive components)