What's the difference of a high-pressure mercury vapor light bulb and a standard mercury vapor light bulb? Could you run a high pressure mercury vapor light bulb on a standard mercury vapor ballast?

Both are two different names of the same lamp.

The earliest designs of mercury lamps were medium pressure, then the high pressure (MB) versions were developed, which were more versatile, smaller and could operate in any position. The original medium pressure (MA) lamps could only be operated vertically base up, although they could be run horizontally with a magnetic arc deflector.

They are electrically identical, so can be run on the same gear.