Is this used for a single, specific occasion (I swim on Friday = this coming Friday); a single, indefinite occasion (I swim on a Friday = on some Friday or other but I am not say when); for a regular habit (I swim on Fridays = every Friday); or more than one of the above?
The first one. It's not necessarily this coming Friday, but it's clear between the speakers which Friday they're talking about.
It might be used in the second meaning as well, but the phrase "egy pénteki napon" is a better translation for that.
"on Fridays" = "péntekenként"
So if it is a kind of arrangement, why "I am swimming on Friday" is not accepted?
You can always add some modifiers to the sentence "Pénteken úszom", and it will sound more of a regular habit. Every - Minden, usually - általában, ordinarily - rendszerint, etc.
I think "péntekenként" can be thought of as the equivalent of "cumaları", for your Turkish mind.