rs232 (db9) pinout question

Hello everybody. How's going?

I have a very old equipment here that uses the RS232/DB9 port and it's connected to a old pc. The equipment is very expensive, so we can't replace it for a new one, and we need to replace the pc for a new one. But new pcs doesn't have a DB9 port so we need to rely on DB9 to USB adapters.

The thing is our pc tech said there are two different RS232 pinouts. And I don't know if this makes any sense because old pcs just came with a RS232 port and that's it, you just connect the cable and install the equipment software without worrying on pinout, but I'm not a pc expert so I may be wrong.

So, is he correct and indeed there are two different RS232 pinouts and I should worry about getting the correct adapter, or is there only one pinout and and adapter will work?