The requirements do not demand an open-water swim, though that's implied. The requirements do not demand a child use a certain stroke. The requirements also don't explicitly demand that the mile swim be continuous. So everything we see here, including doing the swim in stages, meets the letter of the requirements. So, yes, if a lap pool swim is okay with the aquatics counselor, then it's okay.
Yes, there's a difference between swimming a mile in a pool versus swimming a mile in open water. But there is a far bigger difference between swimming a mile freestyle versus swimming a mile using a resting stroke. A boy who swims freestyle for one mile continuously in a pool is a far better swimmer and athlete than a boy who backstrokes and floats his way to a mile in open water. Remember Mao's swim in the Yangtze? 15 kilometers in one hour doing the breast stroke -- eight minute miles! That was an open water swim. But I don't think Mao earned the one mile swim badge. Current did all the work.
I say whatever you decide a one-mile swim should be for your kid, go for it. The requirements are deliberately vague and open to interpretation. That's done on purpose. You, as a scoutmaster, merit badge counselor, or parent, are the person to set the standards. If you set low standards, your scout learns or achieves nothing except rank advancement or a patch. If you set high standards, then your child scout, achieves, and develops, even if they fall short of earning the patch. It doesn't matter what other people choose for their kids. What you choose for your kid is what matters.
For my scout, I decided that a one mile freestyle-only swim in a lap pool with no gliding after flip turns (so swimming the entire distance) and no stopping is a high achievement for a 12 year old, worthy of the mile swim badge. Sure, it's not as challenging as an open water freestyle swim in deep water. However, if he backstroked or doggy-paddled on open water for one mile, I wouldn't let him have the badge. I don't consider resting strokes -- or incapable strokes -- to be swimming. But that's the standard I chose for my scout. I'm perfectly okay with different choices that people make and I'm happy with the reasons they make those choices. We just differ in our criteria. Folks who think the badge should require an open water swim, they're thinking about outdoor competence, survival, and skills. Great. Those who are okay with resting strokes, dog paddling, or staged swimming, they're thinking about developing character, self-esteem, and confidence. Great. I, on the other hand, interpret the badge as a continuous freestyle with no stops and no resting strokes. Pool or open water, I don't care. I'm thinking athletics, strength, stamina, fitness, and performance. Different set of criteria entirely.