Live comedy is often called stand up. What would you call a comedy video not performed to an audience? This could be characters performing a story, someone talking to a camera, or someone more abstract.
The critical aspect of stand-up comedy is that it is a direct communication between the comedian and the audience. The performer breaks the fourth wall in stand-up. It is often in the form of a monologue, but can involve a byplay between the comic and the crowd.
While stand-up is often (perhaps usually) performed in front of a live audience, there is no reason it cannot be performed to a broadcast only audience or even filmed or taped and replayed later. Many comedians relish the feedback of a live audience, but audio recordings of stand-up routines were a staple in the 1960s (Bob Newhart, The First Family, etc.)
Stand-up is distinct from a comedic skit in which the players interact with each other but not with the audience, such as situation comedies. Skits that act out humorous scenes and ignore the audience are not truly stand-up. Some vaudeville routines that involve a dialog between two or more comics are borderline stand-up in that they may or may not treat the audience as being present.
Stand-up also can be distinguished from slapstick comedy in which the performer often falls down.
monologue noun a : soliloquy 2 b : a dramatic sketch performed by one actor c : the routine of a stand-up comic
The comedian is famous for his monologue about winning the lottery.
I would still call it "Stand-Up Comedy", since it would still be a solo standing comedy performance, just without a 'live' audience. Though the audience is not present, the existence of an audience can be inferred by the style of performance, so long as they are still 'addressing an audience' in some way as a solo performer.
It may be a bit unusual, since the reaction of the audience is usually seen as a key part in the performance, but it can still be done and still be considered stand-up.
To differentiate between the two, you could call it a "stand-up recording" or a "recorded act" to indicate that it was not performed live, or just call it a 'non-live stand-up performance'.
It would be called a shoot, [studio] shot, or [studio] taping. If you are in marketing you might call it a live taping or just use the term live. Everything is "live" but we tend to take that as people can't do retakes.