There are different strains of flus, colds, and recently, a few different strains of coronavirus.
You can't generally be reinfected by the same strain. What can happen is that new strains can infect you. There are tons of cold strains around, and the flu mutates quickly enough that each year we get a new one, so vaccines have to be redeveloped for each new round of mutations.
Coronavirus is the same way, except that as far as I know, it mutates much more slowly than cold and flu viruses, meaning that vaccines (and having gotten it once) will be equally and more effective than flu or cold vaccines.
Unless the word "generally" is doing a lot of work in this sentence - YES YOU CAN be reinfected by the same strain. It's called "homologous reinfection": the person's natural immune response is not always up to the task of actually providing immunity to a specific viral strain, and that exact same viral strain can reinfect the patient even immediately after recovery.
This idea that the human immune response is always 100% effective (assuming the patient survives) is frankly a bizarre misconception: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3863957/
He is immune. Similarly, we say that cars can cross the Golden Gate Bridge, despite the fact that the bridge could be destroyed (by bombing for example) or that one could call the NASA crawler-transporter a "car" that is 90 feet wide. Absurd, exotic, and rare possibilities don't change the basic truth.
Vaccines do work like getting the illness, or less so. The illness is nearly certain to provide long-term immunity, but a vaccine is not quite so likely. Those who can't generate immunity will survive the vaccine but not the illness.
It's more likely that the reinfections came from different immune responses, where the first infection was beat with a cytotoxic T-cell response (viral defense) but not an antibody response (viral and bacterial defense), the latter of which increases immunity with antibodies.
The re-infection cases that I read about were shown to have been from different strains. Perhaps this is not the common case, but without more information, I am reluctant to accept your statement at face value.
This has not, to my knowledge, happened yet.
> I don't believe any sensible person believes that they become immune to all strains after recovering from one, and I do not believe that Trump was saying this. Do you?
I certainly do believe that the message trump intends to convey is that he is immune to the coronavirus and can no longer be re-infected period.
Second of all, this is not how vaccines work. Recovery from an illness does not mean that your immune system was maximally effective: a weak immune response coupled with anti-viral medication and hospital treatment could mean that your immune system is not equipped to fight the virus a second time. This happens often with flus and colds.