Wrong isn't the word ide use use, but it's good to understand how everything works for when you can't get by with just a push.

The suction force is noticeably stronger when you get the knack of it. If you start with a push you are more likely to compact the clog and make it more difficult to unstick. Try starting with a good strong pull. My rule of thumb is is it's slowly draining, I be very careful not to send a push down when first putting the plunger in sand getting the air out of the cup

It's again not wrong, and depending on what went down there, it might need that push to break it free, but best practices makes your life easier when it matters, not when it doesn't.

Prepare for the worst type thing.

Other note, a common problem people have seems to be not having enough water in the bowl for the plunger to be doing much good. You can just add it from a sink if you don't want to risk a full flush(or just don't know the toilets flush quantities) and don't want to risk overflow

If it works, it works, I guess. It didn't at our case (clogged too badly, the previous inhabitant did some, weird shit), hence the plumber.

I did it like you did and then the shit water came up the bathtub several cms high. Later attempts changed nothing. Was a disaster. But the plumber went wild and it worked.