I've seen many contests define mobile as something like being able to legally move and broadcast. Well, using an HT while walking I can legally move. Does this count as mobile contacts for a contest?
Insofar as there is an agreed-upon definition of the term “mobile” (and “portable”, which is closely related/confusable and so I'm covering it here), this is how I have most frequently seen the terms used when people are being precise about it:
Portable: A station set up in a temporary location.
If you've parked a radio on a picnic table and strung a wire antenna to a tree, you're operating portable.
In some contexts there may be the additional implication of being a significant distance from your home location.
Mobile: A station which is capable of being operated while in motion.
The most common type of “mobile” is a station installed in a car/truck. For other means of transportation, you usually hear it qualified as “pedestrian mobile”, “bicycle mobile”, “maritime mobile”, etc.
I've never seen anyone say that pedestrian mobile isn't a kind of mobile. So, to the first question you asked, Is using an HT and walking considered mobile, I say “yes”. (And practically, however you're labeled, you've taken on the limitations of a mobile station — small antenna, no earth ground, battery power, etc.)
But in any specific actual contest — if the rules aren't clear, ask for clarification. It's the opinion of the contest organizers that actually matters.