Is the mobile connector you buy from Tesla waterproof?
Waterproof as in being submerged? Definitely not. Waterproof as in being able to resist some rain hitting it if it is properly mounted off the ground? Yes. However, if the rain is very severe (like a hurricane where the water is pouring down like a waterfall or going at high speeds), probably not.
Plenty of people have used it outside in the rain with no problems, although some had have problems and Tesla mentioned they don't cover water intrusion failures (although they did cover "goodwill").
The teardown of the Gen 2 Mobile Connector shows the housing is pretty well sealed (glue/ultrasonic welded so very difficult to take apart), so it fairly resistant against the weather. The points where there may be water ingress that can cause problems is where the adapter plug connects to the unit and also the cable itself (some people say the door open button eventually failed in thread above, although that may not necessarily have to do with water ingress).
This thread has more details. There was someone concerned about holes on the side, which they thought were vent holes that go directly to the electronics, but those don't actually do. Instead they are drainage holes for the adapter plug, which indicates Tesla designed it to handle some rain. The electronics appears to be completely sealed.
Home Charging
As such there's this warning in the manual:
"Warning: If rain falls during charging, do not allow rain water to run along the length of charge cable, causing the electrical outlet or charging port to become wet."
Of course you need a outdoor rated outlet too.