The way I would approach this in a usability test, “says” would be what a user says during the test without any prompts other than being asked to talk through what they are doing.
After the usability test, during the interview phase, we’ll often ask the user to further clarify “what were you thinking when you did/said this.” We also might ask how they felt during a particular task. Technically this is what they “say” but is does take more prompting, and you do have to be careful that your questions aren’t leading to a particular response.
The article you linked seems to imply feeling and thinking as giving more context to what the user says, this could include some interpretation based on tone and body language that aren’t necessarily apparent from the written quote, but there’s a slippery slope here to putting words in the user’s mouth, so I would personally avoid this in most cases. With general emotions, like frustration and boredom, this might be more acceptable, but I wouldn’t want to say that a user felt satisfaction, for instance, without the user specifically expressing that.