While looking for research jobs in industry, I noticed that some companies over both, research positions and postdoc positions. What exactly is a postdoc position in industry and how is it different from a research position that requires a PhD? Is it just that you spend your postdoc years in a company instead of academia?
Post docs in industry do work similar to university post docs, but their research is a property of the company they work for. Typically, a position advertised as an industry post-doc is one that would allow for external publication (i.e., in scientific journals) of the work completed (but if publication is important to you, you should of course verify this before taking such a position).
Post docs in industry are typically fixed-term appointments, like in academia; there may be opportunities for continuing or for promotion within the company, but these aren't guaranteed, and you wouldn't have to be "fired" or "laid off" at the end of your term to be without a job - you would need to check with someone to see how this would affect your eligibility for unemployment benefits, etc, depending on your location.
I would expect that a company that is offering both full research positions and a separate post-doc program is expecting that applicants for the full research positions will have either relevant post-doctoral experience or industry experience: their post-doc program is probably intended for recent PhD graduates.
Lastly, although these generalities may help you, there is no legal definition of a post doc for these purposes, so you should be prepared to ask questions about everything that is important to you before accepting a position.