grammar - For the first time - the first time

I would like to know the difference between "For the first time - the first time". It seems to me that sometimes they are interchangeable but at other times they are not.

For example: (not interchangeable)

1) Most animals can swim right away the first time they go into water.

2) Most animals can swim right away for the first time they go into water. (seems like they might not be able to swim the second time)

For example: (interchangeable)

3) They met for the first time in 1980.

4) The first time they met was in 1980.

Plus here is an explanation by a Londoner

*|No, they mean different things and have different grammar. 'For the first time' is a complete description of a time, and it implies other times that came before it.

For the first time, he won a prize. [He had never won a prize before this. This was his first time to win a prize. But he might have tried many times before.]

'The first time' needs to be followed by a relative clause saying what is 'first' about it. Then, including the relative clause, it forms a complete description of a time. It also implies there were not other times before this - this one is the first.

The first time (that) he took part in the competition, he won a prize. [He had never taken part in the competition before this. And when finally he did take part, he won.] |*

Is the explanation correct? I have made up two sentence to test the explanation. Would it be correct to say that in 5 it is necessary to understand that I had been trying to win a prize before yesterday?

5) Yesterday I won a prize for the first time.

6) The first time I won a prize was yesterday.