The logout page shouldn’t even be presented, it’s an action only page. Not sure what you mean about register.
I took screenshots for both at https://imgur.com/a/cuOINju
The logout page should not be directly linked to. It needs to have a nonce generated and appended to it. If using nav menus, use the TML actions. Otherwise, you’d need to use wp_logout_url() in PHP.
Your registration page looks fine to me: https://imgur.com/a/HPoeH0R
That’s the coming soon page. The registration looks like https://imgur.com/6TihwPi
I’m already using TML Actions for the logout. https://imgur.com/a/UIaEtmr
I don’t see that register page.
Do the logout links have a nonce appended?
The site is under construction which is why I sent a screenshot.
The login does have the nonce. The top left says “Something went wrong” https://imgur.com/a/LkACBvH
Sorry, I can’t help troubleshoot that one if I can’t see it. My guess is your theme is targeting only text inputs like (e.g. input[type="text"]), whereas the email field is an email field (e.g. input[type="email"]).
As for the logout issue, this is likely caching. Determine if your host is utilizing server-side caching, or if you are using a caching plugin. If so, have all of TML’s paths excluded.
You cannot tell by https://imgur.com/6TihwPi what the problem is? It looks like it’s just a css issue.