"I think the possibility of having your location triangulated while moving defeats the purpose of not letting it be triangulated only when stationary"
From what I've read, the majority of the time cellular triangulation is used against a person, it's used in real time against stationary targets, not moving ones. Stingray devices, which also perform triangulation, are usually mounted in vehicles, such as a surveillance van. Kingfish devices are a portable version that can perform the same function.
Not only is triangulation easier against a stationary target, it's also more useful against a stationary target, because when you're stationary, you're usually at a location connected to your identity, such as your residence, or your job.
Knowledge of only one location by itself might be sufficient to identify you, but knowledge of multiple locations would identify you with nearly 100% certainty.
"The amount of time spent at a place could still be known"
They amount of time spent at a location would be known, but the location itself? Not really, unless it was already known in advance, in which case triangulation wouldn't even be necessary to begin with.