Two (or more actors) connected to one use case mean that all actors are involved in this use case. If you want to show, that either one of multiple actors can be involved, a generalization is a good possibility.
The notion of "primary" actor is not specified in UML. Therefore, I talked about actors involved in a use case. Of course, in the description of the use case you will be clear about which actor initiates the use. This will be the primary actor.
All the problems you have with «extend» can be solved by simply omitting it. In my opinion «extend» doesn't contribute to a better model. If you want to say that there is an overlap in the description of several use cases, you could use «include». But actually, you should not worry too much about these relationships. After all, you want to describe what actors are using the system for. Each actor goal is an own use case.