My first thought will say to replace the battery cables. If you have power from the battery outside the vehicle (external tester), then you only have a few options when starting. If engine is turning over consistently, then the starter is engaging. I would start cheap and replace battery cables. You could still have starter issues, but don't replace more than necessary, and since wiggling the cables does make some difference and they were corroded badly, then that would be my first step. It could be just poorly connnecting battery cable ends (due to corrosion), but replacement of the cables is not too expensive or hard.

Thanks,
Jim