I'm looking for a word

stroking?

"To pat" is to touch quickly and gently.

How about "caress"?

Different words describe similar actions in different circumstances. The action is pretty much the same whether you are petting a dog, stroking a fur coat, or caressing your lover. However, if you talk about petting your lover or caressing your dog, people will look at you very strangely. "Stroke" is a more neutral word; you could use it in any of these situations.

"Pet, stroke and ca

I can imagine using the word 'rub' to describe the way someone might touch their belly after eating.
But 'rub' would not work well to describe how you might touch the face of someone you love.

Perhaps stroke is the word I'm looking for. Is it often used when you "stroke the face of a baby" and"stroke the face of your lover"? Or should we use rub?

I think "stroke" is probably the word you want. You can gently stroke someone's face. "Rub" sounds a little more energetic -- you rub lotion into your skin. You rub yourself with a towel after a shower.