Full Idea
An 'informal proof' is not in any particular proof system. One may use any rule of proof that is 'sufficiently obvious', and there is quite a lot of ordinary English in the proof, explaining what is going on at each step.
Gist of Idea
An 'informal proof' is in no particular system, and uses obvious steps and some ordinary English
Source
David Bostock (Intermediate Logic [1997], 8.1)
Book Reference
Bostock,David: 'Intermediate Logic' [OUP 1997], p.327