What is the status of the verb "booster"? (1) Should we say "Get boosted" or "Get boostered"? and (2) Its use in AmE meaning "to promote"

There are different ways of interpreting this question, which will lead to different answers to it.

Purely prescriptivist approach

Verb booster does not exist, because it does not appear in reputable dictionaries.

Purely descriptivist approach

Quite a few examples of such use of the word can be found (some of which are quoted in the question), including in professionally edited publications, so obviously the word is a part of the language.

Middle-of-the-road approach

While the examples of booster as a verb exist, this use is not well established, and one may wonder whether it is a good idea to emulate these examples, and so make the use better established, or treat it as something that it would be better to discourage.

Now, the first thing to notice is that the uncontroversial noun booster is derived from the verb boost. That verb is well established and its use is uncontroversial. Given that we have it, what reason is there to use booster as a verb? The answer is none, as long if it is used in the contexts in which boost could be used. Some of the examples in the question are like that. There is no reason to speak of boostering self-confidence/capacity/immunity/shoreline defenses/moral intention, when one could have spoken of boosting these things.

However, in the contexts that have to do with vaccinations, boost needs something like immunity as its object. One cannot say that a person has been boosted, if one means that the person's immunity has been boosted by a booster dose of a vaccine. We thus need a different verb if we want some term for such a person to be its object. Using booster as a verb is convenient for that purpose.

There is thus a good reason to embrace booster as a verb in the sense of administer a booster dose of a vaccine to somebody, but no good reasons to use it in the cases in which boost can be used.

(The use of booster as a verb may also be justified in the relatively rare cases in which its meaning is promote a cause in the manner typical of a booster, in the sense of the noun booster that is confined to the U.S., and probably has a somewhat old-fashioned ring even there.)