Are there any clear-but halachot as to what makes somebody your ‘Rabbi’, specifically for the Din against calling ‘your Rabbi’ by his first name.
I live in a very tiny Jewish community where there is no Rabbi, only laypersons.
The closest Rabbi is 200 miles away, and I occasionally consult him for Halachic guidance and general advice, but we do not have week-to-week contact, and only see each other perhaps once or twice a year.
Then there are Rabbis whose shiurim I have listened to every day for years (specifically, a daf yomi shiur), but I’m assuming that without any physical or person connection that can’t be a ‘Rabbi’.
According to the strict halachah, the prohibition of calling a rabbi by his first name applies to one’s “rav muvhak” (the rabbi from whom one learned most of one’s wisdom), and to known great scholars in Torah and righteousness (Yoreh De’ah 142:30; 144:10).
However, today, one may not refer to any rabbi (meaning, a rabbi worthy of the title…) by first name alone. Today, such a reference would be disrespectful, just like it is not respectful to call a (secular) judge by his first name alone.
For sources, see at length Yabia Omer, Vol. 1, Yoreh De’ah 18.