Come to think of it, I would change the definition of the Present Perfect to "an action already completed BY (not AT) the present time, or an action begun earlier and spanning a period UP TO the present time."

1. This would cover:
(a) I have seen Simon = BY NOW, I have seen Simon. ("Seeing" of Simon is done, completed, finished, over by now.) This also distinguishes it from the Past Perfect = By then, I had seen Simon.
(b) I have lived here since 2002 = From 2001 TILL NOW, I have lived here. ("Living here" is ongoing, not finished yet, not over yet, up to now.)

2. Using "already" in the definitions for the Present Perfect and the Past Perfect is useful because it means "by now" and "by then".

3. If the sentence uses words like "since" or "for", it still fits the Present Perfect "rule", because both words mean "from an earlier time till NOW (or then, if past perfect)" = over a PERIOD, not a "past time".
(a) I have seen Simon FOR the past two weeks = saw Simon a number of times, from two weeks ago till now.
(b) I have lived here SINCE 2001 = lived continuously from 2001 till now.

4. (a) Yesterday, I have seen him. (not OK)
(b) Yesterday, I saw him. (OK)

5. (a) I have seen him for the past two weeks = I saw him (a number of times) over the past two weeks (from two weeks ago till now). (OK)
(b) I saw him for the past two weeks. If meaning = I visited him (a number of times) over the past two weeks (from two weeks ago till now). (OK)

6. (a) I have seen him already. (OK)
(b) I saw him already. (not OK)
(c) I see him already. (not OK)

7. (a) I have seen him before (= previously). (OK)
(b) I saw him before (= previously). (not OK)
(c) I see him before. (not OK)

8. (a) I have seen him before this. (OK)
(b) I saw him before this. (OK)
(c) I see him before this. (not OK)

IMO, (6) and (7) are the occasions why we need to have the Present Perfect.

In 7, the adverb "before" modifies "have seen". In 8, it modifies "this". So 8 b "I saw him before this" = "I saw him (at a past time)", therefore Simple Past here is OK.