Being "presidential" really does mean everything that it connotes. Indeed, the connotation of being "presidential" is being professional and to act with professional decorum.
The site, freedictionary.com states the meaning as an adjective:
Adj. presidential, relating to a president, or presidency; presidential; befitting a president; looking presidential; statesmanlike.
Indeed, the whole idea of being "presidential" is to act like a statesperson.  Indeed, a president absolutely must act as his own foreign dignitary and it is absolutely a requirement of the job that such a person act with a sense of dignity and respect. In general, the American People usually require it to be a "MUST HAVE" as part of the job requirement.
This means that such a person would have to "act with class" without exception and to be professional at all times.  It also requires that a such a person not use profanity, invoke "locker room humor", be crass or be abrasive and reviling in behavior. I was just 13 years old when Ronald Reagan took office.