What about "drop smoking, leave off smoking, give up smoking"? Do you use them when you quit smoking a next time? 😁

You can GIVE UP SMOKING, not the others.

We also say KICK THE HABIT

It seems to me that "kick the habit" looks like our "завязать с привычкой".

I don’t know, but it generally refers to cigarette smoking in English.

"завязать с чем-либо" is informal and normally used with bad habits including smoking.

E.g. Я завязал с пьянкой. — I gave up drinking.

There you go!

Uly, what about "break smoking" and "stop smoking"? Are they in use?

and "quit smoking"?

QUIT and STOP, but not BREAK