"At home" or "home"

Solution 1:

I haven't researched this, but it seems to me that at is

• optional when home is the complement of a form of be (am, was, are, been etc)

• required when home is a modifier (as in your last example)

• disallowed with go, come and synonyms.

But I'm sure there are some exceptions to these rules of thumb. One interesting one is your third example:

He came to my place at 6pm and by that time I had already been (at) home.

where at changes the meaning. Had already been home invokes the idiomatic meaning of been as a sort of verb of motion, and implies that I had been home and gone away again.

Had already been at home is barely possible for me without an expression of duration (Had already been at home for several minutes).