is it true you have to show that you are vaccinated to travel internationally?

I was not asked for proof when I flew from Phoenix to Seattle on Southwest, but masks are mandatory.

Some countries now require proof of vacination and or negative COVID test prior to travel. Each one is different

I do not believe you do. You may have to have proof of a negative test to return though. I know you have to test negative to return from Mexico but not sure about other countries.