The B18A1 came in the '90-'93, and the B18B1 came in the '94-'01 Integras.

They most likely swapped it out because it the B1 was blown... Or maybe it was stolen and the motor was yanked and they replaced it with the A1.

Is/was the motor converted to OBD2 to match the car, or is it OBD1?

The B18C is the JDM GSR. The USDM GSR is the B18C1, that came in the '94-'01 GSR Integra. There is also a B17A1 that is VTEC equipped that came in the '92-'93 GSR's.

A B18C1 (GSR) ECU will work, but it's not needed. It's a VTEC ECU (P72 is the ECU code), and the B18A1 and B18B1 is non VTEC. If you were to get a P72, it won't do nothing, and it would have to be remapped to work properly with your non VTEC motor. You also can't remap an OBD2 ECU, just OBD1, which is found in the years from '92-'95, and before that was OBD0.

I hope that covers it for you.