I have 2 identical overhead projectors with bad diodes, but the diodes have different numbers. How do I find the right one that works?
One diode has failed shorted. A diode reduces the RMS voltage by a factor of \$\sqrt 2\$ minus the diode drop, so 120VAC becomes around 84VAC.

The current is about 5A, so to be safe the diode should be rated well over that (turn-on surge) and will require a large-ish heatsink. 

I am not sure what is happening with the second lamp-24V does not make a lot of sense. If it appears to be of normal brightness, that could simply be the way your meter responds to half-wave rectified AC. 

I am not convinced that replacing the part with either OEM part is ideal. At the very least the diode will be likely to fail when the lamp burns out. 

Edit: your diodes are 6A axial lead general purpose types rated for 1KV. They are marginal- the NTE one has probably been replaced once already- NTE parts are typically used by service technicians, not manufacturers. 

You can buy 10A types in the same package also for cheap (dimes) but I suspect that heat is a big problem. What I would do is graft in a $3 35A 1kv bridge using only one of the 4 17A diodes (that's important- use it as a bridge and the lamp will blow) and screw it to the case or other heatsink. The bridge will have spade terminals, mounting hole and adequate insulation (but safety needs to be considered- earthed chassis only). I am not going to try to re-engineer your situation remotely without photos etc, but that is what I would try to do- and also avoid that supplier in the future.