When you get that link preview as you have shown, when you pass your mouse over it, there will be a down-arrow appear to the upper-right of it. Click that and choose Convert to Plain Link.
Initially, I though this was due to Substitutions > {Smart Copy/Paste, Smart Links} being selected, but disabling either or both does not change the behavior of having the link converted to the Link Preview when pasted into the compose windows. This must be a new feature of Mail in Ventura 13.3.1 as I see no Mail settings to disable this behavior.
I saw no obvious resources in the Mail defaults dump that could control this link preview badge appearance.
The Link Preview badge survives a trip through gmail and back to Apple Mail.