Misinterpretations of Hilbert's Theorem?

Question

I've seen a few posts here that make certain claims that are related to Hilbert's theorem. For instance:

"I know that there is no complete surface embedded in $\Bbb R^3$ of constant curvature $-k$ for any $k$."

"However, as Hilbert showed us, the reverse is not true; we cannot embed the hyperbolic plane into Euclidean 3-space."

"$H^2$ does not isometrically embed in $R^3$ (Hilbert's theorem)."

However, these should only apply to smooth ($C^\infty$) embeddings. Nash-Kuiper, particularly, guarantees the existence of an isometric $C^1$ embedding, since the Klein disk model is a short map from $H^2 \to \Bbb R^3$.

This was noted in the answer to this MO question.

There's also this answer to the same question, which says there's no $C^2$ embedding.

But then on the other hand, this page on hyperbolic crochet at Cornell claims that the crochet models are not $C^1$ and can be extended indefinitely, which would appear to violate Hilbert's theorem.

At this point, I'm very confused about what is and isn't allowed regarding isometric embeddings of $H^2$ into $\Bbb R^3$. Obviously no infinitely differentiable isometric embedding exists, but there are obviously $C^1$ embeddings of the whole space. Furthermore, there are also isometric embeddings of compact subsets of $H^2$ into $R^3$, such as the crochet models, and it's not clear what differentiability restrictions there are for that.

Is there some way to better understand the slew of somewhat-contradictory statements made above? What, exactly, is and isn't allowed regarding isometric embeddings of $H^2$ into $R^3$?