Una sola persona - una persona sola

Una sola persona - una persona sola
 

El café estaba vacío. Había una sola persona.
El café estaba vacío. Había una persona sola.

I find this a little confusing. I think that the second one means there was a person sitting down at a table who was unaccompanied and definitely not with another person; but does the first one not mean the same thing?

Many thanks in advance.

Both mean a single person. The first has 'single' meaning unaccompanied (alone); the second, unattached (not married).

I agree with Rusty, but depending on the context, there may be a nuance in meaning:
- Había una sola persona. -> There was only one client in the coffee shop.
- Había una persona sola. -> There was only one person and they had no company; they were sitting alone.