Can I unlock the Crystal Cruiser with Type C of the Rock Cruiser?

I can confirm that the Rock Cruiser Type C can bypass the first two steps of this quest chain, finally having done it in game. You must find the beacon event in the Rock Homeworlds sector without the aid of a quest marker, but the event can still be generated irrelevant of the stasis pod quests, and your beginning crystal crew member will give you the blue choice option once the event is found. Long-ranged scanners help a lot with this, as the event should only be present at a beacon with no ship detected, as well as no store, no distress call, and no hazard. Once you enter the hidden sector, you do get a quest marker for the crystal ship location, which is quite nice.

I guess I should have read the updated wiki page that I linked in my question before posting.

Alternatively, the Rock Cruiser type C in the Advanced Edition starts with a single Crystal aboard the ship, allowing you to skip the first two events normally required in order to allow access to the blue option when you find the Ancient Device. This method requires completion of the other two achievements (or repeating this achievement) due to needing the Rock Cruiser type B. Although it still requires luck to find the Rock Homeworlds and to unlock both the Type B and Type C ships, it requires arguably much less luck than the previous method.

I'll be testing it out later on to be sure, but I'm guessing that got added to the wiki page because someone already confirmed it :)

Yes you can use the Rock Cruiser Type C's crystal member. And even better, if you name the crystal crewmember "Ruwen" you get a quest marker instead of a normal beacon when you enter the Rock Homeworld so you don't risk missing it.