"Check with" or "check in with"

"check" in the meaning: to look at something or ask somebody to find out if something/somebody is present, correct or true or if something is how you think it is
(source: Oxford Learner's Dictionaries)

Check with your doctor before beginning any exercise programe.
You'd better check with Jane what time she's expecting us tonight.

"check in" [US] to find out or report the status of someone or something especially as part of a regular pattern of communication or monitoring (often followed by on or with)
(source: Merriam-Webster)

I'm just checking in with you to find out how your doctor's appointment went.

https://youtu.be/RECIiC5phwU?t=221 (3:41)
https://youtu.be/1qq7lDL-bzY?t=397 (6:37)

What is the difference between the two verbs in these situations? Are there any nuances? In what cases "check in with" used and in what ones "check with"?

"Check with" is verifying or consultating something/someone, before doing/saying something.

"Check in with" is more of a nicety; letting someone know that you are (aware that they are) still there. Or, in your example, following up on something.