Is vomit an excretion?
Solution 1:
If you read on in the page you link to, under the medical definitions, you would have found:
[2. b.] a waste product (as urine, feces, or vomit) eliminated from an animal body : excrement—not used technically.
Now, the definition above gives a good definition of the technical use:
[2. a.] something eliminated by the process of excretion that is composed chiefly of urine or sweat in mammals including humans and of comparable materials in other animals, characteristically includes products of protein degradation (as urea or uric acid), usually differs from ordinary bodily secretions by lacking any further utility to the organism that produces it, and is distinguished from waste materials (as feces) that have merely passed into or through the alimentary canal without being incorporated into the body proper
So, in a technical sense, vomit is not an excretion. In a more general sense used more informally in medicine (by which I suspect it may relate to the fact that cleaning it up is part of the more informal side of medicine) vomit is an excretion.
In general though, excretion (and more often excrement) is commonly used of faeces, often as a euphemism, while those who studied biology at a secondary-education level might well know the technical sense too. Neither understanding matches vomit. So while you can point to a dictionary and argue you're within the letter of a definition, it's not a good choice of word. (By all means send readers to a dictionary when the word you settle on is clearly the best choice, but that's not the case here).
Perhaps "bodily functions" would be a better choice for tag; it covers the same territory, and is less likely to meet objections around definitions.
Solution 2:
I think discharge is probably a better generic word for it, since (as you point out) excretion has specific biological meanings.
dictionary.com says:
discharge (noun)
(25.) something sent forth or emitted.
free dictionary's medical section also has this to say:
a setting free, or liberation.
matter or force set free.
an excretion or substance evacuated.
Solution 3:
Excretion has the sense of an externalization of something on a regular basis, such as when the organism is functioning routinely.
Discharge seems to suggest the organism excluding something on an extraordinary basis, that is, when the organism is functioning on an urgent basis. It is used more often when describing the body in stress (infection, nausea).
Vomit seems to better fit discharge than excretion.
I do not think "intention" is a criteria that can be used to distinguish between vomit and urine. The body "intends" to eliminate both of them and the conscious mind may or may not have voluntary control over these processes.