But if you take the typical idea of immunity as "immune to this disease once you've had it" then catching covid does not make you immune at all. With some people catching it within 3 months or less of a previous infection

It's literally the same as with the common cold or with the flu. I'm not claiming that reinfections can't have bad consequences, I'm claiming reinfections are less common if you've been infected recently, which is kinda the definition of immunity (reduced likelihood of being infected after overcoming a previous infection). "Completely immune to the disease once you had it" works for some diseases like measles, but it doesn't for others like flu or common cold, or COVID, that doesn't mean there isn't an immunity boost after infection.