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Undocumented in this context typically means they lack the necessary visas or documents to legally be present or work in the U.S

That's rather ridiculous terminology, isn't it? It's not about the documents, but about the permission the documents represent. It's like calling theft "undocumented retrieval of items", because I lack the documents to prove those items are mine - even when documents do exist, that prove those items aren't mine.
The terminology came about because people didn't like them being called "illegals".
Are you sure? There is a subset of illegal immigrants that walked across the border and talked to nobody. No asylum waiver with a border patrol person, nothing.

That group is entirely unknown and undocumented to the Federal government.

I dont think its a reference to any other kind of immigrant.

It’s a synecdoche.

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