Using the Rogue Scout's Skirmisher feature, which states:

You can move up to half your speed as a reaction when an enemy ends its turn within 5 feet of you.

Additionally:

Being Prone, PHB pg 190/191

Standing up takes more effort; doing so costs an amount of movement
equal to half your speed. For example, if your speed is 30 feet, you
must spend 15 feet of movement to stand up.

And on a relevant topic, Does standing up from prone trigger the damage from Booming Blade?. General Consensus: No.

Can I spend this movement to stand up from being prone?

The concern here is that, while standing up from being Prone costs movement, is the movement still valid, considering that standing up from Prone is not considered "movement" for the sake of other effects that rely on actual movement/distance (like Booming Blade)?

Arguably, the rogue may stand up

The character is not literally "granted" a resource that is called movement but that is in fact the way the rules of movement talk about how it is used:

However you're moving, you deduct the distance of each part of your move from your speed until it is used up or until you are done moving.

In practice, movement is a resource that may be "deducted from" or "spent" (see Being Prone) and which can have "cost" penalties (per Difficult Terrain and crawling in Being Prone). Additionally, the Normal and Skirmisher movements are granted/refreshed using the same terminology:

Normal movement rule: On your turn, you can move a distance up to your speed.

Skirmisher movement rule: You can move up to half your speed as a reaction

So if the two types of movement are obtained/granted using the same "you can move" phrasing and movement is talked about as a resource that can be spent independently of whatever physical space may be traversed (e.g. over difficult terrain), it stands to reason that the movement associated with the Skirmisher feat may be spent to stand up from being prone the same way a character can with normal movement.

In the related question about Booming Blade, the selected answer makes this same point. It is possible to use the movement resource for something other than directly moving. To wit: one "spends" 10 movement to move 5 physical feet through difficult terrain. It is because of this ability to use movement (the implied resource) without moving that Booming Blade does not trigger in response to standing up.

In short, if you subscribe to this interpretation, where the phrase "you can move" is functionally identical to "you are granted movement," and that there is a movement resource that is independent of a physical translation across the battlefield, then standing up is permitted in this situation.

Yes

As you mentioned, to stand up:

[you spend] an amount of movement equal to half your speed.

The rules for prone movement (including standing up) are listed under Movement and Position in the Player's Handbook, right between moving through Difficult Terrain and Moving Around Other Creatures. In my interpretation, I see no reason to disqualify standing up from being a type of movement.

For the Rogue's Skirmisher feature, since you can move up to half your speed, you can simply spend the whole of that movement allocation to stand up.

No

The Skirmisher feature says:

You can move up to half your speed as a reaction ...

Standing up is not moving per the rules' definition, you just have to spend movement to do it.

Standing up takes more effort; doing so costs an amount of movement equal to half your speed.

Since standing up is not moving, then you cannot use the Skirmisher ability to do so.