Also noticed I got emails about the warnings but the inbox didn't change state to indicate that, maybe because I warned myself.

If you send a notification to yourself I think it would auto mark read.

"The main issue is that the reason for the warning links to the post the user was warned for rather than the explanation providing a link to the offending content."
This sentence doesn't make sense to me. Aren't "reason"/"explanation" and "post"/"offending content" synonyms?

The reason appears under the warning tab, the email shows the post and doesn't show the warning, infraction details.

For this, would you want me to do similar to how I did it when I used to run my former website? I refactored the warning module and made hooks for each type of punitive action. Each hook had a function for applying the action, reversing the action, the form field to use on the warning form, and what text to use on the warnings tab and in the sent private message.

Sounds good :)! The main thing is for it to be user-friendly for everyone involved. I think others than me are likely best positioned to determine what that means.
I would add though that it does need to be clear what text is only for staff to see and what is for the warned member to see.
I would also say it needs to not become any harder to warn a member.

Noted. Functionally from a UI perspective, it will work and look the same (aside from adding on this issue of warning messages including text explaining the punitive actions taken). Only difference is instead of everything hard-coded in the module, each type of punitive action and its text will be a hook. This also allows for extending the warnings module easily in the future for other types of punitive actions based on what modules are installed if desired.