My take on this is a tad different:
1) In my experience, the term "still" or "still alarm" refers to an initial dispatch of a response (which may or may not be limited to a single Company or piece of apparatus), usually initiated by Fire Alarm, based on something other than the authority of a struck box.
2) Where a Company itself encounters a situation requiring a response (either by itself or by others), the term I've usually encountered is "on site." (E.g., "Engine 1 to Fire Alarm: onsited an MVA Smith and Main Streets, probably PI.") The result of an "onsite" report may be he intitiation of a "still," but this doesn't equate "still" with an encountered situation.
3) In the jurisdictions I'm familiar with, the term used for situations where a person comes up to a Company and advises of a situation requiring response, which the Company itself hasn't witnessed, is "citizen report," versus "still," precisely to negate any implication that the Company is reporting its own observation.
The origin of the term "still" is a bit unclear (at least to me), but an explanation I've heard more than once, and which makes sense, is that the response has been ordered even though the alarm bells that historically would be sounding if a street box had been pulled are not ringing, i.e., they are "still."