Why do we say i definitely have no friends, but not i definitely was not at home?
have is also auxillary verb.

Because have in "I definitely have no friends" is used as main verb and not as auxiliary/helping verb. And, "I definitely was not at home" is more correct than "I was definitely not at home"

Explanation:

Usual position of adverb definitely (which is an adverb of certainty) is after auxiliary/helping verb and before main verb.

"I definitely was not at home" is more correct than "I was definitely not at home". The first sentence conveys a sense that I was not at home, with stress on I  and no clue or stress on wherever I was. However, if the speaker wants to put more stress on location/place "not at home but somewhere else" then second sentence ("I was definitely not at home") is better. It conveys I was not at home but somewhere else.

the answer is a).