Present perfect or simple past, which one to use when a specific time is not mentioned?

Question

2. Is there a specific time mentioned?

The Simple Past is used when the time is CLEAR:
They met on Sunday. My birthday was last week. John started his business after he graduated. (We know exactly when.)

The Present Perfect is used when the time is NOT SPECIFIC:
They have met already. I have celebrated my 20th birthday. John has started his business. (We don't know exactly when.)

I found the above info here and in some other books.

I concluded that there is a difference between using simple past and present perfect when the exact time is not specified, i.e. using simple past in this case is not considered correct form. The reason I concluded this way is that if they were interchangeably usable in the case of indefinite time in past, it should have been mentioned both tenses are usable. But there is not any such thing here.

Is this conclusion wrong, according to American English?

Answer 1

There's a lot of confusion around this, and a lot of half-true "rules" are taught.

The actual rule for the present perfect is fairly simple. The present perfect is a present tense, designating a present situation, so it cannot be used with a temporal expression which does not include the present, the moment at which the sentence is uttered.

Consequently:

^okThe directors have met is acceptable, because there is no temporal modifier. This merely asserts that the consequences of the fact of their meeting (whatever those consequences are) are now in play.

^okThe directors have met this week is acceptable, because the temporal modifier this week designates a timespan which includes the present: it runs from the beginning of the current week right up to the moment of speaking.

^∗ The directors have met last week is not acceptable, because the temporal expression last week designates a timespan which lies entirely in the past, ending with the beginning of the current week.

Note, however, that an entirely past temporal expression may be included in a present perfect sentence if it is 'bracketed' with commas or dashes or parentheses; it is then understood as a 'supplement', not integrated into the sentence but added to the sentence as an afterthought.

^okThe directors have met—just last week, as a matter of fact.

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ADDED:
DamkerngT points out that you conclude that use of simple past with an indefinite temporal expression is incorrect. This is not the case. The simple past can be used with no temporal or with indefinite temporals:

^okThe directors met.
^okThe directors met sometime.
^okThe directors met before lunch.
^okThe directors met after that.

The only sort of temporal which cannot be used with the simple past is one which cannot be interpreted as lying entirely in the past, such as now or currently or tomorrow.