I use close and deep third-person POV synonymously.
I have a rule of thumb for deep third-person POV:
Don't do anything that makes the reader see the POV character from the outside.
Therefore, things like "a brilliant smile" as in "I smiled a brilliant smile" are out. Unless you're in front of a mirror or looking at yourself on a video recording it sounds off...
You can use the "I"-test to figure out if your deep POV is working or not.
However, that doesn't work with "his father". "My father" sounds just like something you could be thinking, so "his father" passes the "I"-test.
However, at least to me, "his father" creates a picture of the character, his father, and their relationship. At a distance.
"His father" doesn't pass the "don't make the reader see the POV character" test.
Of course, when your deep POV character is thinking about someone else, it's ok, even needed to make a visual picture, but you should avoid doing so for the POV character.
So, what to use instead?
What does Ray use when thinking about his father? Father, dad, pop? Use that instead.
Take:
His father was always late for work.
...and turn it into...
Father/dad/pop was always late for work.
That doesn't create an image that includes the POV character.
You also get a bonus in that it's almost as if it was the POV character's thought, but without having to get awkward with altered perspective or tense for thoughts. ("My dad is always...")
This also works regardless of if your perspective shifts into plural deep POV (?) or stays in singular. "Father" is "Father" regardless of if it's "their father" or "his father".
This brings me to another rule of thumb I use for deep POV: Remove as many possessive pronouns ("his", "my", "their"...) as possible from the text. (In fact, I think you could safely remove them all!)
And while you're at it, try to limit the use of all the other pronouns as well.
E.g.
He picked up the old, heavy book. It was dusty. The pages were so frail they rustled when he turned them over. Finally, he found chapter ten. It contained a description of ...
Could be turned into.
The book was dusty and heavy, the pages so frail they rustled. Chapter ten contained a description of ...
I say "try to remove pronouns" because this one can get awkward if you apply it in a hundred percent of the cases. (Or maybe it's just extremely hard to do but done successfully will create the next great novel?)