Learn English – the difference between “serious” and “self-serious”

The meanings of jargon terms often have essentially nothing to do with the meanings of the English words they're made from. Nowhere is this more the case than in mathematics.

I would use whichever term seems better established—regardless of whether it sounds artificial to native English speakers who aren't mathematicians—so as to give the reader the best possible chance to figure out what I'm talking about.

Updated: I should just answer your question. To my ear, there is a verb filter, and a count noun filter. There is also a non-count noun filtration. (You can count coffee filters, but the filtration of water through coffee grounds isn't something you can count.) So already the mathematical use of filtration as a count noun ("a filtration") differs from the everyday use.

I am pretty sure I never heard the verb filtrate used in everyday English until I started searching for such uses just now. A Google search for filtrated hits mainly dictionary sites. At the moment, the first non-dictionary hit is a link to this question! I can confirm that to my ear, it's hardly a word, and it sounds weird and artificial. Filtered sounds much nicer to me. It is an actual common, everyday word (and, correspondingly, gets hundreds of times as many Google hits).

Notice that in your own examples, you vouched for a person, but guaranteed an action. This is exactly how I'd use these words, and it does seem to reinforce your intuition: "vouch" is about a person's character; "guarantee" is about actions or things. We might also say, however, that "guarantee" emphasizes the future, while "vouch" emphasizes the past. To explain my position, I'll call on the Oxford English Dictionary.

Guarantee

The OED lists the following meanings for the verb "guarantee":

1. To be a guarantee, warrant, or surety for; spec. to undertake with respect to (a contract, the performance of a legal act, etc.) that it shall be duly carried out; to make oneself responsible for the genuineness of (an article); hence, to assure the existence or persistence of; to set on a secure basis. (Example: "Written languages guarantee a systematic pronunciation.")
2. To engage to do something; to warrant or ensure that something will happen or has happened. (Example: "I'll guarantee that he'll never return to Bengurry.")
3. To secure the possession of (something) to a person, etc. (Example: "Christ guarantees to the faith of His brethren...a true quittance and defence from sin.")
4. To secure (a person or thing) against or from (risk, injury, etc.); to secure in (the possession of anything). (Example: "Angus was strongly disposed to make the attempt, if he could be guaranteed from loss.")

All of these definitions and examples state that something will happen in the future, or promise to do something or cause something to happen. In the first example, "Written languages guarantee a systematic pronunciation," the claim as I understand it is not that written language can't exist unless a systematic pronunciation is already in place; the claim is that written language tends to create a systematic pronunciation because we come to associate certain symbols with certain sounds. (Obviously, this claim would not apply to all writing systems.)

"Guarantee" is also often applied to products ("We guarantee this vacuum cleaner for life"); again, this is a statement about the future: the vacuum cleaner will not break down, and if it does, the manufacturer will replace it.

Vouch

"Vouch" has quite a few meanings, many of them archaic and/or specific to the legal realm. "Vouch for", however, has only these three:

1. To speak or bear witness in behalf of (a person); to be surety or sponsor for. (Example: "I dispute not the lad's qualities, for which your reverence vouches.")
2. To supply evidence or assurance of (some fact). (Example: "The certainty of the law...is vouched for...by the results of experiment.")
3. To give personal assurance of the truth or accuracy of (a statement or fact). Also with accuracy, truth, etc., as object. (Example: "I love the country better than ever, I can vouch for that.")

I take it that the first meaning is the one you had in mind. My belief that "vouch" emphasizes the past, rather than the present, rests on the fact that even though you vouch for a person with an eye to the future--securing a job for her, for example--you're vouching for certain qualities, like character, experience, expertise, or personality, which are already possessed by that person. You're not guaranteeing that you'll take over the job if she fails; you're just saying that she has the expertise and background necessary to succeed at it.