“Climb high, sleep low” is a strategy for climbing really big mountains “expedition style.” Climbers “climb high” to fix ropes, carry loads to the next camp, etc. Exercise at higher elevations helps them acclimatize. Then they return to a lower elevation (usually the highest established camp) to “sleep low.” The next day, they typically move camp to the higher location and repeat the process.

If you wanted to apply “climb high, sleep low” to Whitney, you would do something like this:
1. Carry some gear to Outpost Camp and return to the portal to sleep
2. Hike to Outpost Camp and sleep
3. Carry some gear to Trail Camp and return to Outpost Camp to sleep
4. Move camp to Trail Camp and sleep
5. Climb to the summit and return to Trail Camp. Return to the portal with your gear the same day or the next day.

Sleeping in Lone Pine isn’t a strategy for acclimatization. It just un-does whatever acclimatization you accomplished at Cottonwood Lakes. As the article you referenced says, the goal is to ascend “more than 1,000 feet in a day.” It is not to sleep at the lowest elevation you can find and then climb to 14,500’ the next day. If you want to acclimatize, you sleep at the highest altitude at which you don’t get sick. And you climb even higher than that during the day.