Regarding covid spike proteins: covid itself does have a very prominent and distinctive spike protein.
Most (possibly all) of the covid vaccines have used qualities of the spike protein to 'describe' covid to your body, so that your body knows to respond aggressively tow covid infection. Once the body is aware of the threat from covid (either through a vaccine or actually getting covid), it makes antibodies against a future covid infection.
Finding covid's spike protein in someone means that they currently have or are recovering from a covid infection. Finding covid's spike protein in someone does not mean anything about their vaccination status.
Finding antibodies to covid's spike protein in someone means that their body has been trained to react aggressively against covid. This could have happened because they got covid, or were vaccinated, or both.
Though none of them are quite as distinctive as the spike protein, covid does have other proteins in it as well - for example, it has a fairly stable nucleocapsid protein. And if you get a covid infection, your body will produce antibodies to covid's nucleocapsid protein.
So far (at least that I'm aware of) there are no covid vaccines focused around the nucleocapsid protein, which would also make the body produce covid nucleocapsid antibodies. So if you find nucleocapsid antibodies in someone, that person has had a covid infection. The antibodies fade over time, so their infection was probably within the past year or so.
And finally, the vaccine is injected into your arm. The vaccine stays in your arm. Your body 'looks' at the vaccine, sees it as a threat, and produces antibodies that spread throughout your body. Which is what we want it to do - after all, what use would a vaccine be if it only worked in one square inch of your shoulder?
To summarize: Finding covid spike proteins means a covid infection has recently happened. Finding covid spike antibodies means the body has been infected, vaccinated, or both. Finding covid nucleocapsid antibodies means that a covid infection has happened within (roughly) the past year. And the vaccine stays in the shoulder; the antibodies the body makes in response to the vaccine spread throughout the body.