Hi Efi, welcome and I hope you get lots of useful advice as a result of this post. Working in a small company like you do can give you lots of opportunities to get involved and it sounds like you are getting a good spread of work and there is scope to expand it, potentially, where you are.
I have to say I find job titles can vary from company to company and depending on the sector. Some businesses or organisations have more structured, linear and hierarchical career progression and in others it's about broadening the role and the structure can be quite flat.
I'd suggest looking at other companies and their job descriptions. There are sites out there that would allow you to compare and contrast roles and that might help.
I'm sure you are looking at the CIPD professional wheel (I know it's not called that but hopefully you know what I mean) so you can see what areas to get experience of.
Personally, I don't see much difference between your role and a coordinator. I think an HR generalist covers the whole employee lifecycle so is broader than a specialist role which might focus purely on employee relations for example.
Hope that helps and adds to your thoughts and you get lots of other insights too. Good luck with your HR career.