What would you call somebody that has a strong way with words, especially in terms of public speaking?
An orator. The definition is a public speaker, particularly one who is skilled and forceful with their words.
There is elocutionist also.
a public speaker trained in voice production and gesture and delivery
And a little passage from the history of speech:
The success of lyceums attests to the value Americans placed on elocution and oratory early in the 19th century. In keeping with this emphasis a number of individuals began providing elocution lessons for those interested in improving their own ways of speaking, reading aloud, giving oral presentations, or singing.
Elocutionist was the name given to both those who performed orations themselves and those who taught others how to perform. These US specialists in oral presentation called themselves elocutionists, following the already established elocution movement in the United Kingdom.
Well the professional job they would have would be announcer or host and in the broadest sense, politician.
In clubs there is often a toastmaster.
For the literary types you could use rhetorician.