Well, this is interesting. It's one way of defining the difference between feelings and emotions (fellow clinical psychologist here)

I do find when doing a CBT mood log with a patient, spending time with an emotions list (as many as 90 words, none of which would fit your definition of "feelings") can be very helpful.

STEP 1: EVENT: Joe insulted me

STEP 2: NEGATIVE EMOTION:

"I feel that Joe shouldn't have done that."

no, that's not an emotion.

"I feel that he's bad for insulting me."

Ok, try to find one word to describe your emotion.

"Judgmental"

No, here's a list: "annoyed, frustrated, angry, hurt, anxious" just pick one word

"Angry"

BINGO!

Now I like the definition of emotion that it is generally a mental (NOT analytic!) interpretation of a sensation.

Then what is a feeling?

Here I find the Buddhist definition the simplest: a value imposed on sensations of pleasant, painful or neutral.

But then there's a whole area we've lost. Nowadays the positive psychologists are talking about "positive emotions" but I like to refer to THOSE as deeper feelings. Hope that's not too confusing:

Love, compassion, empathy, kindness, gratitude, gratefulness, generosity, etc.