[Horsa]

You are right that there is a time relationship here, but I am afraid your rule doesn't quite work. Your first two examples use 'remember' in its present form. However, if we use the past form we can say

"I remembered to buy the book."

and this is very definitely in the past.

The relationship between "remember" + Gerund and "remember" + infinitive is better explained thus -

1. "I remember seeing the book in many bookshops."

What happens first in this sentence "remember" or "seeing"?

Answer: "seeing" happens before "remember"


2. "Remember to buy the book."

What happens first in this sentence "remember" or "to see"?

Answer: "remember" happens first because if I don't remember to do it I won't buy the book.

Then looking again at the example I gave earlier -

3. "I remembered to buy the book."

What happens first in this sentence "remember" or "to see"?

Answer: As before when we used the infinitive "remember" happens first.