Recently I presented part of my thesis to an international conference. This is the most original and pioneering part of my thesis.

The question is 'is there a commitment for researchers to publish their work in conference proceedings, or this is optional'? Is it rude not to publish in the conference proceedings after presenting at the conference? What do you think?
For instance, I may select to publish the same paper in an academic journal instead... or as a book, when I publish my thesis. If yes, do I have to email the conference committee and explain that I will be publishing my paper somewhere else? What if I offer to publish in the conference proceedings a note (i.e. a very short paper) on the topic instead?
Thanks in advance
Mara Sp.

No you aren't expected to (and don't have to notify anyone). If you think it can become a journal article then I wouldn't!

What do you mean by publish "in the conference proceedings?" Most conferences will only publish an abstract, which would not prevent you publishing elsewhere as well! (BTW- not sure what field you are in).

I am doing law in combination with human rights. We publish the whole paper in the conference proceedings.

It is perfectly acceptable not to publish in the proceedings, and I don't think it's necessary to send any follow-up emails. If it's been a very large conference, some editors might ask the authors in advance to state whether or not they plan to send in their contribution (this helps them organise the volume's sections, ensure sufficient number of reviewers, etc.).

Regarding other formats (notes,commentaries, etc.), these may be mentioned in the CfP the organisers/editors will send out, but if not, it might be worth contacting them and checking if they would be willing to consider such contribution.

I'm not sure what the status is in places like the US and the UK, but here in Australia, conference proceedings are no longer being considered towards your publication count.

If this is the same situation in the US/UK etc, I would actually avoid publishing in conference proceedings and instead incorporate the feedback from your presentation into your thesis and subsequent publications of that thesis (if you go journal article route). It's now not just how much you publish, but rather, how much of that is in high level journals.