I tried the space junkyard start. It has the same issue with specializations that I came across with one of the other non-story starts. When you select your first spec, a message says you'll be able to unlock an additional spec at levels 12, 18, and 25. What happens is, you jump straight to level 14 almost immediately, but there is still no way to unlock another spec.

I have played up to level 20 on the non-story start, and still both "upgrade spec" and "select a second spec" say "wait."