Okay:
T.I.E. Fighter: Used in A New Hope, the white one
TIE/ln (look closely, it's a L N, but it's not capitalized): Upgraded post yavin design, Ln stands for Line Fighter
TIE/In (I N this time): Tie Interceptor
Guns? 2 Lasers 4 Blasters:
The original production model only utilized four laser cannons, one on each wingtip. The cannon hardpoints located on the ball cockpit remained, and two additional cannons could be placed in those hardpoints if room for additional power generators could be found.
Later, Imperial scientists reengineered the TIE/In so that it had two laser cannons and replaced the original wingtip cannons with blaster cannons, making six cannons altogether. The two empty hardpoints under the cockpit were utilized in this model, creating a more powerful TIE/In. It is believed these models participated in the Battle of Endor.
and of course evidence to back it up:

So we'll have two variants, one with 4 lasers, one with 2 lasers and 4 blasters.