I'm fond of "attending".

I'm haven't read Gendlin but got my interaction with emotions from other sources.

When trying to explain it to someone I think it can be useful to teach by example. "Where in your body do you feel the emotion? Put your hand on that spot."

The hand is a good feedback to know that the person understood what you want from them. It also helps them to be more aware of the emotion.

From there it depends on what I want to do. If the goal is simply about knowledge it can be useful to let the person describe what they are feeling.

I don't know whether having a word to describe the process helps for implementation in a way where it becomes your default way of dealing with emotions.

"Monitoring"? (I'm not actually familiar with the subject.)