Hi medic31,
Thank you for posting in the Apple Support Communities. I see you're running into an issue with the reading list in Safari.
It sounds like you're able to save a website or article to the reading list, but you don't see it when you tap the book icon followed by the reading list (glasses) icon; is that right? Add webpages to a Reading List
You mentioned checking some settings; to clarify, which settings did you work with already?
Does it help if you Restart your iPhone, iPad, or iPod touch and then try to add a website to the reading list as a test?
I also notice you're running iOS 12.1.3, and updating is a great way to clear up unexpected behavior in some cases. Here's How to back up your iPhone, iPad, and iPod touch and Update your iPhone, iPad, or iPod touch.
I hope this helps get you started. Kind regards.