Transgender male: Doctor said "It's a girl!" The person said "Nope, I'm a boy!"
Transgender female: Doctor said "It's a boy!" The person said "Nope, I'm a girl!"

Trans guys can be called guys. Trans girls can be called girls. So yes, unless they ask you to, you don't have to tack "trans" on the front of someone's gender.

Also, like others before me have said, don't say "was male at birth" or "used to be a boy" or "turned into a girl" or anything like that. Instead, try to adopt phrases like "assigned male at birth", "designated gender/sex", and "transitioned from sex1 to sex2" though even that last one can be improved upon.