There doesn't seem to a word which is an exact parallel and addresses all these cases and is in common usage, or even uncommon usage. Your phrase "an obvious example of" may be one of the best ways of conveying your precise meaning, though there are alternatives which would be much more natural in common English but aren't identical in their connotations.
As @Glorfindel says, typical is plausible and common for the first case:
He/she is a typical rude person.
But one would rarely say "That was a typical insult" as the implication would be that it is typical of a person to be insulting, rather than that the statement was a typical example of an insult or that amongst insults the statement was typical. And "he is typical health" would not work; "he is typically healthy" is sound but suggests that the person is usually healthy, with the possible connotation that sometimes they are not.
Picture would be reasonably common for the second case though it is now quite cliche:
He is the picture of health.
It conveys that someone is obviously healthy, and you can tell by looking at them. But less so for your other cases. "He is the picture of a rude person" would be an unusual statement, since people rarely look rude. "That was the picture of an insult" likewise is unhelpful as it is hard to visually picture an insult.
Common usage would suggest clearly or obviously as mildly but not deeply emphatic (unless additional emphasis is placed on the word) and easily understood, but it doesn't have the connotation of being an obvious example, only that it is very apparent that the person has this quality.
He is clearly a rude person.
He is clearly healthy.
That was clearly an insult.
Exceptionally is quite viable and more emphatic, but suggests an abnormal and extreme case compared to the average person, and (as with "clearly") one which can be obviously identified:
He is an exceptionally rude person.
He is exceptionally healthy.
That was exceptionally insulting.
Apparently doesn't really work, since it has an undertone of being dismissive or disbelieving. "He is apparently an idiot" is quite scathing. "He is apparently healthy" suggests he may not be. "That was apparently an insult" is ambigous depending on context, possibly giving the person the benefit of the doubt, or suggesting that the person who thought themselves insulted may be wrong.
In geek culture, canonical would be quite viable for some of your usages, interchangeably with textbook which @Glorfindel suggests (and "textbook" is in wider usage and hence is probably better unless you're writing for a geek audience).
He is the canonical rude person.
That was a canonical insult.
It suggests that there is a (tongue in cheek) canon or doctrine to which may one authorititively refer, which defines such things, and that this fits the criteria.
In popular culture, compare the line from the film Pulp Fiction:
Whether or not what we experienced was an "according to Hoyle" miracle is insigificant.
This likewise suggests a putative canonical authority, in this case, Hoyle and is used for emphasis in a dismissive statement, since it sets up the non-existent canon as an obvious strawman. But this wouldn't fit a serious text or repeated usage in the same text.
