I'm the one who made that comment. I am a native English speaker. I will explain what I was thinking, though in preparing this answer I may have talked myself into accepting that there is no difference in meaning.

Provided that "do one's best" is an idiom, is "I did the best I could" a proper phrase to be used at all?

I don't see why one phrase being an idiom has anything to do with whether another phrase is "proper", but in any case "I did the best I could" is a common expression.

is there really a fine difference between the two?

The two phrases can be used interchangeably. The meaning is very similar, or, arguably, the same: the shared idiomatic meaning is that I tried as hard as I could.

The reason I said that they don't mean quite the same thing is that in my experience when someone says "I did the best I could", what they mean but aren't saying explicitly is "I did the best I could in the circumstances". For example, in preparing dinner I did the best I could with the ingredients available in the refrigerator, but if I'd had time to go to the supermarket I could have bought more ingredients to cook something much nicer. Or in running a race I did the best I could given that I had the flu, but usually I'm a better runner than that.

"I did my best" can be and often is used in exactly the same way: that in the circumstances I could not have done any better. But sometimes the literal meaning is intended, that the result is the best I am capable of in any circumstances - I have come across this intended meaning occasionally. I have never heard "I did the best I could" used to mean the best I'm capable of in any circumstances. So that is the subtle difference that I had in mind. (I accept that not everyone will see any difference between the two. Add I said at the beginning of my answer, I'm not sure that I see a difference anymore either. The two phrases do feel slightly different to me, but I can't explain that any better than what I've already said. Sorry, I did the best I could...)

I'm the one who made that comment. I am a native English speaker. I will explain what I was thinking, though in preparing this answer I may have talked myself into accepting that there is no difference in meaning.

Provided that "do one's best" is an idiom, is "I did the best I could" a proper phrase to be used at all?

I don't see why one phrase being an idiom has anything to do with whether another phrase is "proper", but in any case "I did the best I could" is a common expression.

is there really a fine difference between the two?

The two phrases can be used interchangeably. The meaning is very similar, or, arguably, the same: the shared idiomatic meaning is that I tried as hard as I could.

The reason I said that they don't mean quite the same thing is that in my experience when someone says "I did the best I could", what they mean but aren't saying explicitly is "I did the best I could in the circumstances". For example, in preparing dinner I did the best I could with the ingredients available in the refrigerator, but if I'd had time to go to the supermarket I could have bought more ingredients to cook something much nicer. Or in running a race I did the best I could given that I had the flu, but usually I'm a better runner than that.

"I did my best" can be and often is used in exactly the same way: that in the circumstances I could not have done any better. But sometimes the literal meaning is intended, that the result is the best I am capable of in any circumstances - I have come across this intended meaning occasionally. I have never heard "I did the best I could" used to mean the best I'm capable of in any circumstances. So that is the subtle difference that I had in mind.

I accept that not everyone will see any difference between the two. As I said at the beginning of my answer, I'm not sure that I see a difference anymore either. The two phrases do feel slightly different to me, but I can't explain that any better than what I've already said. Sorry, I did the best I could...