Actually the custom in the real "new kadampa tradition,"
aka the gelugpa, is to call teachers "gen," teacher.

I doesn't matter what people call the person, the titles
are credentials given by either one's lineage as embodied
in a monastery, order or ngakpa gompa. In the Gelugpa
order, to be granted the geshe degree, one has completed
15-20 years of rigorous academic training at a monastic
university associated with a monastery. There have been
examinations both written and oral-- just like in a PhD
program. Few who begin the process complete it. Bonpo
has a similar process for its Geshes. If the person has
taken the courses, but haven't sat for the exam, they don't
get the degree or title. Of course, it is assumed that the
Monk or Nun remains affiliated with their monastery
regardless of whether they remain ordained. They don't
ordinarailly break with the tradition that trained them,
or get themselves excommunicated.

The Sakya, Kagyu and Nyingma orders have a monastic
degree kalled Khenpo/ khenmo. The term can also mean
someone who has been a fully ordained monk/nun for 8-10
years, but normally a khenpo also holds the academic degree
of Khenpo. Each tradition is a little different in terms of how
much retreat time is required for practice as distinct from
academic study (memorizing texts and debating). Again if you
don't sit for the exams, the monastery doesn't grant the degree
or title khenpo. I know several Lamas who refrained from sitting
for exams because they didn't want the title. It was a matter of
wanting to maintain humility.

Full ordination in a monastery typically includes a teaching credential.
Few monks and nuns are fully ordained. Full ordination is granted
to those who are sen by their teacher as qualified to hold the vows.
Fully ordained monks without academic credential are called Gelong,
fully ordained nuns Gelongma. Novice nuns are called Ani.

Ngakpa Gomapas are a different animal altogether, but they also have
academic programs associated with their dratsangs (dharma colleges).

The title Rinpoche normally is given to an enthroned Tulku or recognised
incarnation. There are incarnations that are recognized but not enthroned,
usually they are given a letter of recognition but not always. Rinpoche can
also be used colloqually as an honorific for one's own teacher regardless
of whether they are a Rinpoche or not. My own teacher is known to be a
Rinpoche, even though he was never fully recognized or entroned as
a Rinpoche.

The term Lama means "nothing higher-- father and mother" and is
equivalent to the sankrit term guru. Anybody claining to be a lama can
have their credentials verified.