Use the easiest screen for each job
A phone or computer is usually more convenient for searching and comparing films. Find your title here, check its year and choose a viewing option. Then open the corresponding supported app on your television.
Check the edition after switching devices
The television might open a different language option from the one you saw on your phone. Check audio and subtitles before everyone settles down. For a sequel, compare the full chapter name too.
What about casting or consoles?
Those options depend on the viewing service and device. A console app may be supported where a built-in television browser is not. Casting and screen mirroring are also different features, so use the method documented by the service you selected.
My List remains in the browser where you saved it. You can keep it on your phone as a simple list of what to find in the television app, or use backup to move the catalog list to another browser.