Apparently I have a proclivity for long answers, but I thought I'd respond given the viewership on this question.
We can whittle your question down to a more general form: Can my subjective experience be categorized? The answer to that basic question is of course! You made a categorization of how you felt in that moment, and you concluded that you had a feeling of "being extremely present."
But why did you make this categorization? And would others agree with you on it? Would others understand what you mean when you talk about the feeling of being present? Would they make the same categorization?
In brief, you categorized your internal state on the basis of (1) conceptual knowledge that you had accumulated from past experience combined with (2) your perception of the present situation (e.g., Barsalou, 2009; also here). With this information, you cohered all of those disparate and ambiguous elements of your subjective experience into a single concept of "feeling present." And now you're able to reflect on it more easily, act on it more easily, and talk to us about it more easily. Pretty cool!
The Anger Concept
To understand this a bit better, consider a prominent emotion theory (Barrett, 2014), which holds that the sociocultural knowledge we accumulate about emotion categories (e.g., anger, fear) allows us to categorize our own experiences, categorize others' experiences, and communicate those experiences. So as you learn about the concept of "anger," for example, you develop the capacity to categorize your angry experiences ("I'm angry"), others' angry experiences ("She's angry"), and communicate those angry experiences (see, e.g., Caron, Caron, & Myers, 1985; Widen & Russell, 2008). Over time, you learn to make this anger categorization automatically and often unconsciously (cf. Barsalou, 2009).
We know that in many countries, anger is a concept that most people would understand (although even then, there may be differences in definition and understanding; e.g., for the word "happiness": Oishi et al., 2013). However, it's not clear whether most would understand your concept of "feeling extremely present." After all, it's not a traditional Western idea, nor does it fit into a traditional taxonomy of feeling states. Moreover, it's not entirely clear that you can communicate its contents effectively (which is true for many feelings we have).
We can look at research that has operationalized feelings of presence and compare them to yours. For example, one well-known study defined feeling present as a focus on momentary experience (Farb et al., 2007), which they validated in an fMRI study. Would the experience they defined match up with yours?
Streams not Boxes
I can't really speak to whether the feeling you described has a name or is well-known. However, the important thing to note is that subjective experience isn't carved up neatly into boxes. It only seems that way because humans have a natural tendency to constantly categorize things (see again, Barsalou, 2009), which, as you can imagine, is very useful.
Instead, we might describe subjective experience as a "continuous stream." Anywhere along that stream, we can make some arbitrary division. "This part of my subjective experience, and all related instances, will be called anger." "I'll categorize this part as feeling present." And so on. There are virtually infinite ways of categorizing your subjective experience, and so we can't possibly come up with names for all of them. But as you've done, we can try our hardest to communicate our feelings, and most of the time, it won't be in vain!