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:
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