Precise means that on a GC cycle, only true pointers to heap allocated objects are identified as roots, which means that an unreferenced object cannot be kept alive by a value that happens to have a similar bit pattern as a heap pointer. This guarantees that unreferenced objects are eventually cleaned up, but not that this cleanup happens immediately, certainly not as part of a busy loop. (Unless the system runs out of memory, but in that case, a GC iteration is required anyway.)
Since reference counting is GC, isn't reference counting a form of precise GC? That's certainly the scenario I had in mind reading what OP wrote where deallocation within a hot loop would be possible.
Tracing is the technique to find all alive objects from roots. You dereference a pointer to find there some object, and then to recursively iterate over pointers in that object. But before that you need to choose what to use as roots, and here is a difference: you can be precise with choosing the roots, or to err on a conservative side, choosing more roots than you actually need.
This classification was invented decades ago. I think in 1970s or even earlier. Garbage collection is a large field with lots of ideas and research behind it. There are different approaches to structure the heap, to find roots, to trace living objects. These different approaches oftentimes (but not always) replaceable, you can change how you find roots while leaving other things intact.
GC is a large field, it has its own terminology, and instead of asking "why are we inventing terminology", I'd ask "who we are to change the existing terminology".
Precise means that on a GC cycle, only true pointers to heap allocated objects are identified as roots, which means that an unreferenced object cannot be kept alive by a value that happens to have a similar bit pattern as a heap pointer. This guarantees that unreferenced objects are eventually cleaned up, but not that this cleanup happens immediately, certainly not as part of a busy loop. (Unless the system runs out of memory, but in that case, a GC iteration is required anyway.)