The practical challenge of choosing a word seems to be relevant because most words hint at particular side of the subject-object spectrum. I'm not sure if this is unintentional or part of the teaching...
Sheesh. I had a longer post typed in, but I guess it really just comes down the nuance of how words imply a particular style of observer.
Attending conveys both awareness and participation, both subject and object, like "attending a football game". As in both "attending with" and "attending to"? Hokai did you intend this?
Gazing seem very objective to me -- maybe because it's usually use in the form "gazing at" -- but Haguan it seems like you use it as including a sense of absorbtion, implying a subjective effect.
Grok seems a bit intellectual, because it isn't just what is occuring but also implies an secondhand interpretation and knowing... implying an active role for an observer.
Noticing implies a noticer doing the noticing, whereas noting implies a greater focus on what's noted. (to me anyway)
More and more I think I'm of the spectulative/deluded temperment because these thoughts can trip me up so easily during meditation! I've never felt comfortable with what I'm doing in part because the emphasis on either subjectivity or objectivity seems appropriate at different times and even moment to moment. Here ends the confessional