Is the “half your speed” used in standing from prone your normal speed, or current speed?

At my last session, a scenario arose in which a character was attempting to cover ground as quickly as possible, but was knocked prone. On their next turn, they proceeded to stand up and Dash. This led to a moment of confusion while I attempted to figure out exactly how prone and Dash interact.

The player commented offhand that standing from prone halves your movement and Dash doubles it, so the two cancel and they had 30ft to move that turn. Reading the rules, however, I interpreted things differently.

Dash

When you take the Dash action, you gain extra movement for the current turn. The increase equals your speed, after applying any modifiers. With a speed of 30 feet, for example, you can move up to 60 feet on your turn if you dash.

Any increase or decrease to your speed changes this additional movement by the same amount. If your speed of 30 feet is reduced to 15 feet, for instance, you can move up to 30 feet this turn if you dash.

(PHB p192)

Being Prone

[…] You can drop prone without using any of your speed. 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.

(PHB p190)

My interpretation is that your speed is constant – say, 30ft – and your movement is a separate quantity representing distance you can cover this turn. You start the turn with available movement equal to your speed, and then Dash adds another quantity to your movement, while standing up removes a quantity from your movement. Neither affect your speed.

That means that this player starts with 30ft of movement, gains an additional amount equal to their speed (30ft) from Dash, then spends an amount equal to half their speed (30/2 = 15ft) to stand, leaving them with 45ft of movement available for this turn.

In the end, they needed less than 30ft of movement to get into position to attack the bandit leader and the party all made it out alive, so it didn’t affect the outcome of our game, but I’d like to get this straightened out for the future.

Is my interpretation of the interaction between speed, movement, the prone condition and the Dash action correct?