No, standing from prone doesn't trigger booming blade's damage.
I don't see any specific rules definitions for "movement." Based on definitions in other editions, my opinion is that a character "moves" when they walk or use some other "special type of movement" such as jumping, flight, swimming, etc: when they move from their original position, rather than just standing up, dismounting, or otherwise reconfiguring their posture but remaining in the same place.
The spell doesn't say "if the target willingly takes any action." Therefore, attacks and environment interactions are probably fine. The spell doesn't force you to remain perfectly stationary like a statue. Given this, it seems clear to me that doing things that use movement but don't change your tactical position don't count.
Some places in PHB190-191 use the phrase "deduct... from your speed" instead of "spending [X] feet of movement." This might be a clearer way of looking at it. Some actions, such as standing up, reduce the amount you can move in the turn even though they are not "movement" for the purposes of triggering spell behavior.
For a useful corollary, it makes sense that mounted movement would still trigger the effect, even though you're not using your own movement or deducting from your speed.
Although it doesn't affect the lack of definition in 5e's rules as written, D&D 4e says "Whenever a creature, an object, or an effect leaves a square to enter another, it is moving." 4e rules don't apply to 5e, but they provide insight into the background of D&D rule design.
Rules designer Jeremy Crawford unofficially clarified via Twitter that it refers to using your movement, as in "I move X feet" - not a generic "move", as in "I scratch my nose":
When booming blade refers to moving, it means movement in the game's normal sense: moving X feet.
Moreover, standing from prone, while it uses movement, does not count as a move for the booming blade spell, as Crawford has also clarified:
Standing up costs movement but moves you nowhere. To move while prone, you crawl or use magic (PH, 191).
The target perceives the "booming energy" but doesn't automatically know the spell's effects.
There is no general rule that spell targets know when they are under the effect of a spell, and the text of booming blade gives no specific rule as such.
All the target can really know is that they are surrounded by "booming energy." How they choose to react to that is probably a matter of their intelligence and experience as adjudicated by the DM. An experienced caster or warrior might know to stay put and wait for the spell effect to pass, whereas a beast might flee in terror for lack of comprehension.
It's worth noting that the identify spell is the canonical way for a caster to ascertain what spell a target may be affected by. Since only certain casters have access to the spell and it generally costs a spell slot to cast, it's unlikely that a target is intended to know the exact details of a booming blade spell cast on them without having access to identify or a comparable feature. Xanathar's Guide to Everything includes optional rules for identifying cast spells using Intelligence (Arcana) checks, so this further implies that knowing the effects when targeted by a spell is not automatic. It's something to be adjudicated based on the context.