It really is confusing, and there is conflict as to what is really happens. I think our best guide is the story itself. For this answer I will be using the anime as a guide. Lastly, please note that this discussion pertains only to Super Saiyan itself, and not Ascended Saiyan, or SS3/4.
Now, Goku himself says, when training Gohan, that it does indeed come from a need, and if needed you can create that need. Though, I'll have to check with the original Japanese, because perhaps this wasn't the best word to use in translation? Considering that he also notes that the pain of loss as an important factor, this leads me to believe that perhaps it is more of an emotional thrust than a direct need per se.
Either way, in my humble opinion you need a mixture of the two (need and power-level.) If I'm not mistaken, no one is shown doing it without both factors.
- Spent much time training in 100x gravity, which undoubtedly boosted his power-level way up.
- He was pushed over the edge by an emotional trust. (The ruthless death of his friend.)
- Out of pride he spends his time in training trying to become a Super Saiyan, just like Goku, including some time in 450x gravity.
- Out of anger, jealousy, and pride he is pushed over the edge.
- Spent much time training with his father in the Hyperbolic Time Chamber.
- He is pushed over the edge by the distraught he feels at the idea of failing his friends.
Now, this is only valid in the act of becoming a Super Saiyan. Later on, once a Saiyan has learned how to control it better, the emotional thrust is lessened (if not removed altogether.) In addition, when higher forms of Super Saiyan are considered, the emotional barrier is lessened even more so, and the emphasis is placed more on power-level, and perhaps skills.
As for other Super Saiyans whose original transformations are not explicitly shown, I leave that up to pure speculation. At least we have three good cases to base our claims on; perhaps the rest just belongs in the imagination.