According to the Tome of Fate, all CSMs get either a legion bolter or a legion bolt pistol with 4 clips of ammo. The Alpha Legion CSM archetype then grants a legion bolter and a legion bolt pistol with 3 clips for each. So this essentially means I can have either two legion bolters (with seven clipss between them) and one legion bolt pistol (with three clips) or one legion bolter (with three clips) and two legion bolt pistols (with seven clips between them) at the start of the game, correct?

Note that this assumes that Legionnaire = Legion and Boltgun = Bolter as per the errata. IOW, both the Legionnaire Bolter and the Legion Boltgun should both be written as Legion Bolter.

Sounds right. I didn't look at the gear too closely, but it sort of makes sense.