Hello friend - While most of what you have written is correct. We aren't an imperial assassin. We are a death cult assassin and therefore point 1+ 2 don't apply to us. 

As per point 3 we are a death cult assassin which is an entirely different character. While some argue with this - sometimes death cult assassins are trained within those collages as a personal favor to an inquisitor. Which is documented within some books. These chaps are also frequent members of an inquisitor's retinue. But more importantly capable of rising in rank and asending into becoming a full inquisitor themselves. Not all death cult members are raving lunatics all their lives.

As for the equipment I personally don't see it as being a case of  "you have an exitus ergo you must be vindicare" - Too me it's a case that "oh I have an exitus - vindicares like to use them" - In this sense I don't think the helmet and weapon really dictates what your training is, just what equipment you like to use. We can probably argue that a death cultist shouldn't be getting such nice toys - but we are an inquisitor after all :)

Perhaps worth searching the forums, theres a good 5 threads on this exact topic 

Thanks for putting all the references, very handy.