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.
And, while I was just the youngest of teens, I could tell Reagan knew how to act - and act with class. After all, he was an actor!
And, yes, without a doubt, you could tell, he knew what the American Presidency called for in terms of professional conduct. Like him, hate him or leave him, no one in America could ever accuse Ronald Reagan of acting poorly in public and without class.
If anything, this President was the embodiment of what it meant to be a person of class and a world class President. No doubt, I was not nearly sophisticated then as I am now. I can tell you, in all the years I saw Reagan on television I cannot remember one single instance where he acted "UNpresidential" or without class. Sure, there may have been gaffs, but, then again, no one is perfect and, as far as I know, everyone on the planet is human (and therefore prone to make mistakes).
If Reagan really is the great REPUBLICAN ICON of the Republican Party, maybe the the party itself and their classless cadre of presidential cadets could take a cue from their icon - and learn what it means to be a consummate professional and a person of great class and what it truly means to be, "presidential".