Learn English – Word for people who eat out in restaurants/eateries

Q: By which I mean, people strategically position themselves so as to acquire money/service/free stuff at the expense of another?

Scrooge would be a good term for this, after the protagonist Ebenezer Scrooge in Charles Dickens' A Christmas Carol, who was initially only interested in acquiring money at the expense of all those around him.

Q: I don't mean extortion (that's too strong a word for what I need). Basically, someone who is a sucker for free/paid for things and capitalises on the opportunity. Is there a word for it?

You could call this kind of person a freeloader or free rider, which is someone who will happily take free things without perhaps paying their fair share. The Yiddish word schnorrer can also be used for a sponger.

All these terms carry somewhat negative overtones.

No, there is no such word. The closest would be "home training" or "home trainer" but I don't think it is extremely common and is likely to get confused with skill training at home or someone who comes to your home to train you in something.