1. invoke_on_fit_task_begin / invoke_on_fit_task_end
“Invoke” clearly means “call the callbacks’ hooks.”
Common in callback/event APIs, and not overloaded like “eval.”

2. dispatch_on_fit_task_begin / dispatch_on_fit_task_end
“Dispatch” suggests sending an event to all registered callbacks.
Fits well if you think of this as dispatching a lifecycle event.

3. notify_on_fit_task_begin / notify_on_fit_task_end
“Notify” suggests informing listeners (the callbacks) that the task began/ended.
Slightly less explicit that you’re calling a method, but still clear.

4. call_on_fit_task_begin / call_on_fit_task_end
Most literal: we are calling the callbacks’ on_fit_task_* hooks.
Very clear, a bit plain.

5. run_on_fit_task_begin / run_on_fit_task_end
“Run” is simple and implies execution of the hooks.
Could be read as “run (something) when fit task begins” rather than 
“run the begin hooks,” so slightly less precise than “invoke” or “call.”

Recommendation: invoke_on_fit_task_begin / invoke_on_fit_task_end or
call_on_fit_task_begin / call_on_fit_task_end are the clearest and least ambiguous.
I’d lean to invoke_ for a public API and call_ if you want maximum literal clarity.