africa – Can I enter Kenya and Tanzania multiple times on single-entry visas?

I will be traveling to Kenya and Tanzania, and will make multiple stops in each country: Home -> Kenya -> Tanzania -> Kenya -> Zanzibar (Tanzania) -> Kenya -> Home. I will not travel outside of those two countries.

I’ve read and heard conflicting reports for whether or not I will need a single-entry or multiple-entry visa for each country. Some websites state that a single entry is fine as long as I stay in the East African community (Kenya, Tanzania, and other countries), e.g.:

Tanzania single-entry tourist visa validity: up to three months. You can re-enter Tanzania on the same visa (providing it is still valid) if coming into Tanzania for a second time from Kenya, Uganda, Rwanda or Burundi.

While this says I can enter a second time, it does not mention a third entry. Some consulate representatives have told me a single-entry visa is fine, some have told me a single-entry visa will not work since I will enter a third time.

Do I need to apply for a multiple-entry visa to Kenya and Tanzania, or will a single-entry visa suffice?