Shown in the image below is an example of an actor that can perform an action and this action branches into several options (none of them HAVE to be performed). My question is whether the arrows coming from "Manage my events" should be "include" or "extend". In other words, is "include" or "extend" more appropriate if I want to depict a use case branching out into other optional use cases?
if you take a look at UML Specification.
Include : An Include relationship specifies that a UseCase contains the behavior defined in another UseCase.
Extend : A relationship from an extending UseCase to an extended UseCase that specifies how and when the behavior defined in the extending UseCase can be inserted into the behavior defined in the extended UseCase.
So your example means that when a user "manage my event", he always (includes) "Update event", "Add event", and "Invite participant".
I guess that you wanted to model the fact that "manage my event", may be extended by "Update event", "Add event", and "Invite participant". If I am right, "Extend" relationship would be better cf below.