but thought it was just a cold
Coronavirus is just a cold, or perhaps more precisely, what people call "a cold" is caused by a coronavirus (not SARS-cov-2 but one of the relatives of it).
Could my immune system be able to fight the virus without needing the antibodies?
The antibody test looks for a "lock and key" with the spike protein. But a real virus is a 3-D shape with multiple surface proteins and other molecules attached. In general an antibody will match to any possible structure the real virus coat can make. For instance, it could match to part of a spike protein and part of another spike protein and/or some area of the surface in between. Note: I am not 100% certain of the shape and dimensions of the surface of SARS-cov-2 because I haven't really looked into it (don't care) but this is how it works in general. An Ab can match to any part of the viral body surface. You could have such antibodies from previous exposure (or this exposure).
So if you don't match to the spike protein, that doesn't mean you don't match to the virus. It CAN mean that, but it doesn't necessarily mean that.
It's also possible that an Ab for some other coronavirus is a sufficient match that it causes an immune response.
It could also mean that you encountered it, but it couldn't take hold because you are not vitamin deficient and your cells just fought it off. It could even be some genetic reason, like a slightly different ACE-2 protein structure that decreases binding with SARS-cov-2.
TL;DR Yes.
but thought it was just a cold
Coronavirus is just a cold, or perhaps more precisely, what people call "a cold" is caused by a coronavirus (not SARS-cov-2 but one of the relatives of it).
Could my immune system be able to fight the virus without needing the antibodies?
The antibody test looks for a "lock and key" with the spike protein. But a real virus is a 3-D shape with multiple surface proteins and other molecules attached. In general an antibody will match to any possible structure the real virus coat can make. For instance, it could match to part of a spike protein and part of another spike protein and/or some area of the surface in between. Note: I am not 100% certain of the shape and dimensions of the surface of SARS-cov-2 because I haven't really looked into it (don't care) but this is how it works in general. An Ab can match to any part of the viral body surface. You could have such antibodies from previous exposure (or this exposure).
So if you don't match to the spike protein, that doesn't mean you don't match to the virus. It CAN mean that, but it doesn't necessarily mean that.
It's also possible that an Ab for some other coronavirus is a sufficient match that it causes an immune response.
It could also mean that you encountered it, but it couldn't take hold because you are not vitamin deficient and your cells just fought it off.