Can a blood test for covid antibodies tell if you had covid in the past, even if you have been vaccinated and had a booster? in other words, can the test tell the difference between vaccine antibodies and antibodies from infection?

The normal screening tests for covid antibodies will pick up those from a wild covid infection. Only specialized testing for vaccine related antibodies will detect these different antibodies & such tests are rarely found outside of research facilities. They are different.

Yes. Vaccine induced antibodies are limited to spike protein. Infection induces antibodies to other viral proteins. These antibodies can be distinguished thus being able to distinguish between vaccine induced antibodies and those from an infection. Get vaccinated.