I'm a beginner, and I'm worried about practicing scales in an inefficient way. Is it recommended for the long term, for the sake of internalizing scale structure, that I memorize where the fingers cross in each scale, or consciously pay attention to each time I cross fingers? Or is it fine if I simply play the scale correctly by reading the fingers off my book, and over time that will internalize the structure just fine? My concern is that by just doing the exercises in my book, I'll get good at the specific exercises, but won't truly master the scale. Is this idea false?
Either is a good start point. But, remember, learning scales is all you're doing. Learning scales. That's not learning music, any more than learning the alphabet is learning to read expressively.
The 'crossover' points you mention won't necessarily come at the same place in real music, so learning them, and where they are and how they work, is basic to scale playing rather than real pieces.
At some point, though, you won't need to look at the dots, as memory will take over, but one main use of knowing scales is to be aware of which notes constitute which keys, facilitating playing a certain piece in a certain key.
There's at least a third way, too, which will eventually become the norm - playing scales without looking at either the dots or the keyboard. That's what to really aim for.
Fingerings will come with repetition either way. As long as you're playing the right notes, the muscle memory will come.
What you should try to focus on is the note names. Don't just play the white keys; play C natural, D natural, E natural, F natural... Focus particularly on the notes that are different in neighboring scales. Being able to internalize a key signature is a necessity in harder material and thinking about it early will make you a much stronger pianist.
Your book is just a starting place. Go ahead and learn the scales there, and as you internalize them, you'll find other ways to practice them. Changing the rhythm, for example, is a common variation in scale practice.
As you play music, you'll also discover many different ways that scales are used, and these will constitute scale practice in themselves. These may also suggest new ways to practice scales on their own.
It is a good enough start, but just by this you indeed won't "master the scale". I'd say let's go a step back and ask why you want to practice scales. Are you usually sight reading music and just want to learn different keys and scales that you might encounter there? Then sounds like reading off the book is perfectly fine and prepares you to do exactly that. Do you want to use scales for improvisation? Then I'd say it's probably best to practice scales in improvisation - that's also a good way to get to know the scale more from a musical perspective rather than just as a finger pattern.
Both are useful, but don't worry about it too much
The difference in focus won't make much of a difference in your progress, but here's my two cents...
It's always a priority to play consciously, and always listen to your sound. Micro breaks of a few seconds help the thing you're practicing "sink in".
With these out of the way...
Focusing on your score will improve your reading skills more.
Focusing on your hands will put a premium on your dexterity.
Eventually, once you become familiar enough with the handwork/fingerwork aspect, you'll be able to focus on both. At some point you'll have the notes memorized, so you'll barely have to pay attention to the reading/recalling part.