The key is the phrase in your "When I was young".
The past employment history in LinkedIn is used several ways:
- To confirm to a potential contact that you are the Carlos Heredia they worked with 10 years ago on that project in city x for company y.
- To be used as a filter by HR searching for potential candidates,
- or to confirm the info on the resume.
I would make it match what you have on the resume/CV. If it wasn't significant enough to include on the resume/CV, then it probably isn't needed in your LinkedIn profile, unless that is how you expect old contacts to find you.
For company name I would probably use: self employed, or independent contractor.