Say there is a very long article, I took it, then cut/shrink/excerpt it to a much shorter version.
What the proper way to express the meaning/action of cut/shrink/excerpt, but in a respectful way?
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Abridge is a very formal word for this:
to shorten by omission of words without sacrifice of sense
However, in most contexts, I would prefer condense, which the linked definition lists as a synonym of abridge. Abridge is more of technical term used in the publishing industry.
The verb I would use in a conversation or email is simply to shorten the article. "Shorten" is a general verb, not specific to text, but it would be perfectly correct and understood from the context.
The verb abridge is specific to shortening text, but you would find it more commonly used as a base of the adjective "abridged" (and sometimes "unabridged") than as a verb in the active voice (see ngrams). It is indeed quite formal as noted in Max's answer.
You can find some other words that can be used based on these words in a thesaurus.
From a clarifying comment under the question:
"I took the article from Wikipedia then cut it into the above shorter version."
If you are taking a long piece of text and reducing to a smaller number of key points, I would say that you are summarizing it.
: to tell in or reduce to a summary
// I would like to take a moment to summarize the facts that I presented earlier.
// He summarized by saying we needed better planning and implementation.
And from summary:
1 : COMPREHENSIVE
especially : covering the main points succinctly
: an abstract, abridgment, or compendium especially of a preceding discourse