Filling out my intent to lien form, should I put my company name, and my personal name, or just one of those?
The claimant is the person or entity performing the work. If you had a contract with the person, it would typically be the contracting party. If you did work personally, without a company affiliation, it would be your personal name. Though I don't know what your role or trade was from this question, in most circumstances, an operating construction company would be the claimant. Individuals would typically appear as claimants when they're laborers or unincorporated businesses.
A simpler way to think about it is who should the check have been written to had you been paid.