Yes, your Tesla Mobile Connector(usually called a UMC, universal mobile connector) would be a Gen2.
Your UMC can only ever draw 32A, and there's also a thing called a Corded Mobile Connector(has a hardwired 14-50 plug!) that could draw 40A from a 14-50 .
There are tons of problems with your install. Connecting the 14-50 in parallel with the HPWC is a significant violation of the NEC. It will guaranteed trip the breaker if you try to use both at the same time. It will not damage either the UMC or the HPWC, but you won't be charging at all on either once the breaker trips. Violation 1: Supplying a 14-50 with cable/breaker good for 60A is not allowed. The receptacle itself isn't rated to handle that much power. Violation 2: EV charging circuits are supposed to be deficated to one device: Violation 3(similar to 2): 14-50's are required to be alone on the circuit. Violation 4: EV outlets are supposed to be GFCI breaker protected.
The circuit for your 14-50 should be dedicated, and at least rated to 40A(to get 32A out of it), and have a GFCI breaker feeding it. If its wired/breakered for 40 instead of 50A, it should be labeled(at the receptacle) as such.
Sadly, UMCs don't power share, so you can't just put a 60A subpanel in place of your HPWC and feed both devices from that. It'd be a major stretch, but you could still do that as long as you specifically decreased the draw of the HPWC to 16A by hand, in the car, whenever the UMC was in use.
Your electrician should go back to electrician school, unless they are really your 'relative that's good with elecricity' . I hesitate to ask what type of conductors are feeding your 60A HPWC. Maybe I shouldn't hesitate, since if they have to be replaced anyway you might as well go whole-hog and put a 100A subpanel in the garage.