Unreal engine is the easiest to modify and is limitless in what you can add to it, which is why it’s one of the most licenced engines in use by developers.
Source is showing it’s age nowadays, it can’t render such large environments like Unreal2 and up, crytek2 or Dunia, I would expect Valve would change that though, as they seem to be adding all the things Source lacks that are in other engines, mass-AI for example, episode 2’s larger environment rendering (But not as big as the other mentioned engines can do).
Source is good for running on computers from years 2002+, crytek2 was made for the next year of computers O_o
Unreal is definately made to compete with other engines and to be lincenced, U3 tried too hard to look amazing (But it’s impressive how low a system it can run on!).
EDIT:
No, calling Source engine Valve engine would be stupid as they named their engine “Source”, Cryengine, crytek engine, what’s the difference?
Quake engine, IDTech2, what’s the difference?