The TL;DR: It’s ambiguous because Paizo flubbed Technical Writing 101 and uses the same word to mean multiple things. I would give it to the player since I don’t think it hurts anything to say the nimble Rogue rolling out of danger also avoids taking any AoOs in the process.
The Stride as part of Nimble Roll (I assume that’s what you meant to reference) will already provoke AoOs normally. Giving the reaction the Move trait is really just a reminder of that not specifying something new. Otherwise (ie. if we normally ignore the traits of subordinate actions), what is the point of the MAP comment in Attack of Opportunity on p.142? If we ignore the traits of that Strike, it’s not an Attack and thus doesn’t interact with MAP to begin with.
Also, check out the text of Desperate Finisher (a Reaction) on p.152:
You throw everything into one last press. Use a single action that you know with the press trait as part of Desperate Finisher. You forgo the ability to use reactions until the start of your next turn.
The relevant text here is the Subordinate Actions section in the sidebar on p.462, which I’ll reproduce below:
An action might allow you to use a simpler action—usually one of the Basic Actions on page 469—in a different circumstance or with different effects. This subordinate action still has its normal traits and effects, but is modified in any ways listed in the larger action. For example, an activity that tells you to Stride up to half your Speed alters the normal distance you can move in a Stride. The Stride would still have the move trait, would still trigger reactions that occur based on movement, and so on. The subordinate action doesn’t gain any of the traits of the larger action unless specified. The action that allows you to use a subordinate action doesn’t require you to spend more actions or reactions to do so; that cost is already factored in.
Using an activity is not the same as using any of its subordinate actions. For example, the quickened condition you get from the haste spell lets you spend an extra action each turn to Stride or Strike, but you couldn’t use the extra action for an activity that includes a Stride or Strike. As another example, if you used an action that specified, “If the next action you use is a Strike,” an activity that includes a Strike wouldn’t count, because the next thing you are doing is starting an activity, not using the Strike basic action.
And then the bit on p.461 that was noted above:
There are four types of actions: single actions, activities, reactions, and free actions.
So Reactions are actions, and actions can include other actions. Great. That Stride in Nimble Roll is the Stride action described on p.471, but only kinda, and you have to do some inferring about whether something that happens “when you Stride” only happens when you spend an action to Stride or any time Stride is invoked. Hence the ambiguity.
To me, the fact that both Nimble Roll and Mobility include the “You can use this with other movement types” call-outs says they are using the same concept of Stride, and Mobility should thus apply to Nimble Roll.
Someone might say to this, “The second paragraph of Subordinate Actions clearly means a subordinate action never triggers things triggered by using that action itself.” Okay, then the Strike as part of AoO cannot critically hit, neither can the Strikes in Twin Slice, Rogues can’t get Sneak Attack on their Opportune Backstab, and a dozen other things. Some things have to be able to trigger on subordinate actions or everything goes wonky. The interpretation of that paragraph I’ve seen used is that anything specifying a sequence (like the example in the paragraph saying “the next action you use”) usually won’t work since the “next action” is the containing action/activity/reaction/free-action, not the one specified. But anything that just applies to the action without regard for when or how the action is performed applies even when it is a subordinate action.
Unfortunately, Paizo uses “action” to mean 2 different things: a sort of currency you spend on your turn to do things, and various things you can do (which includes “Single Actions”, “Activities”, “Reactions”, and “Free Actions” per p.461). These 2 things frequently overlap, and Paizo’s use of “action” often means “an Action [currency] spent performing a specified Action [thing you do]”, but it also often refers to only one or the other of the two meanings. A good example of this confusion is Barbarian’s Mighty Rage, which is a Free Action with the Trigger of “You use the Rage action on your turn” and says:
Use an action that has the rage trait. Alternatively, you can increase the actions of the triggering Rage to 2 to instead use a 2-action activity with the rage trait.
Notice how they use “action” 2 completely different ways there.
They should have better separated the two and avoided using the combined form, in my opinion. “If your next action is a Stride” should have been something like “If the next Action you spend is to Stride”. Better yet, since it’s already pretty game-ified, just replace Action [currency] with “Action Points (AP)” instead. You get 3 AP each turn to spend on actions and activities (another term they are not consistent about using), but those actions can also be taken under special circumstances, such as Nimble Roll, without spending any AP.
Then they could specify that only things saying “spend AP on an action” requires the action to be used on its own, while all other phrasings that don’t mention AP, like “take an action”, apply to all contexts the action is used in, such as it being a subordinate action.