Usage of while: While he was playing, I sat beside him.

We know that “While he was playing, I sat beside him.” is correct.
What about: “While he was playing, I was sitting beside him.” ? Is this sentence 100% correct? If not, how do you express this is standard English?

Yes, your 2nd sentence is fine. In fact, it is the first sentence that is suspect-- I certainly wouldn’t call it correct. Use the progressive for both verbs if both actions are continuing for the same duration.

They’re both fine to me. Whether they did or didn’t have the same duration, they both make sense and are grammatical.

Thanks Mordant for clarification. I was very surprised by MM’s answer (regrading his comments on the first sentence), but I do respect him. Maybe, at the moment of giving my answer, he just overlooked the first one unintentionally. Thanks a lot to both you and Mister Micawber again!

No, I did not overlook it. I stand by my statement. Now, if it read like this-- While he was playing, I sat down beside him– then I would find it more acceptable, since it now clearly reflects an ‘instantaneous’ act.

Thanks to Mister Micawber again for his logic and clarification.

I think the addition of ‘down’ is a miniscule point and also found the first sentence acceptable.

Beeesneees, I agree with you completely. Because, in Betty S. Azar’s book I saw, While I was sleeping, the phone rang. And in explanation she wrote,
“A while-clause describes an activity that was in progress at the time another activity happened.” Basic English Grammar, Betty. S. Azar

However, as I am not a native, I have to listen to and respect all.

Thanks to you a lot.

That’s just what I said! Somebody ain’t listening to somebody.

I think some misunderstanding is going on here!

Mister Micawber wrote (in his first post) “I certainly wouldn’t call it correct.” Now I think he wanted to say, I certainly wouldn’t call it incorrect.
Sir! Is it okay? And it would mean that, you just added the ‘down’ later to mean its a better way to express it …right?

I respect and thank all!

Actually, I forgot I wrote that; what I remember is that I thought it was suspect. I thought ‘down’ improved it, but others don’t, so there we are. I’m moving on.