This close/shallow/deep etc., POV thing always leaves me cold.
The basic type of POV that is chosen (be that First or Second or Third or Omniscient) is chosen because it seems to the writer to be the best POV to use to connect with a reader and get the desired images across.
Once that decision is made, the depth of POV is something- to me at least- that is not even considered.
A POV either works or it doesn't, and depending upon the type of story the POV can either get deeper/close in on the character and their emotions or move out, as dictated by the changing needs of the unfolding story.
Every unfolding story has varying degrees of closeness or deepness or whatever and that is just the way stories - and life - work. As writers, we would naturally close in or focus more on a character's feelings and emotions during a stressful or emotional scene and move out in other less stressful moments - without having to decide in advance to do that.
I may be wrong and I may be alone, but I don't think 'depth of POV' per se is anything to get hung up on.
Focus on the story.
'He saw' 'He realised' etc., has nothing to do with POV- it's filtering, and a bad habit if it's done without thinking.
The following is just the opinion of a hack writer.
The POV for the tale, Omni or 1st or 3rd usually (I've heard of 2nd but can't recall reading any) is chosen based on A) what the writer is comfortable with, and B) the best way to illustrate the story. In my reading, 3rd is the most common, but that doesn't mean it's best, however, readers may take to it easier due to its familiarity.
When considering depth, I like the term
psychic distance, the closer I am to the character, the more intimate / visceral the story becomes. However, depth shouldn't be constant throughout the story, instead it moves in and out depending on what's going on. If you're writing a combat scene, you can't get too deep into your character's head - because that's really not where they are at the moment because they're relying on muscle memory and instinct to stay alive. Conversation is often the opposite - people spend a lot of energy while 'listening', they're analyzing what's being said, considering their reply, and picking up on non verbal cues.
Filtering (she wondred, thought, realized, etc.) increases distance between the reader and the MC, and is something I avoid as much as possible.