"Stop working" vs "stopped to work"

I want to know, is there any difference between "stop + v.(ing)" and "stopped to + v.". These are example sentences.

I stop working for a month. vs I stopped to work for a month.
I stop watching movies. vs I stopped to watch movies.
I stop cooking for a year. vs I stopped to cook.

I read “I like to do (be) something” vs “I like doing (being) something” but I still don't understand what is the difference or when and how to use them.

Both are correct, but they have very different meanings.

I stopped working means I once worked, and now no longer do. I stopped to work means that I once was doing something (unspecified, based on context), and I ceased from doing it so that I could work. The infinitive (to work) here has the meaning of in order to work and so that I could work.

So I stopped cooking means I once cooked, and no longer do, but I stopped to cook means I stopped [doing something] so that I was then able to cook.

The first sentences here, I think, are meant to be in the past simple, not the present:

• I stopped working.
• I stopped watching movies.
• I stopped cooking.

All these sentences mean that I was doing some activity, working, watching or cooking at some point in the past, and then I finished doing it. In these cases, the verb stop is taking another verb as its complement. This verb must be in the -ing form.

The second sentence in each of the Original Poster's examples uses the verb stop without a complement. Because the verb stopped doesn't have a complement, we don't know what activity was happening. We don't know what I stopped doing. The I stopped part of the sentence is the same as just saying:

• I stopped.

The second parts of these sentences use infinitives of purpose. We use infinitives of purpose to explain why someone does, did, or is going to do something:

• I go to the gym to stay fit.
• I went to the bar to get a drink.
• I'm flying to Paris to visit my sister.

Why did I go to the gym? To stay fit.

In the Original Poster's example sentences, we don't know what I was doing, but we know why I stopped doing it:

• to work for a month
• to watch movies
• to cook

These infinitives are not complements of the verb stop. The are adjuncts (or 'adverbials') which explain why I stopped doing whatever I was doing. They provide extra information.

Hope this helps