Otherwise, a generator that lags in phase becomes a motor.
The grid time is itself an interesting concept. Because the grid runs at a known frequency, it's possible to use it as a time base for a clock. The grid time is considered to be the current time for such a clock. It basically integrates the grid frequency error over time.
Regarding generators that lag the grid, it's not quite as simple as generators turning into motors. The generators all have real time governors that adjust their throttles based on their current operating speed. (The operating speed is directly linked to their operating frequency, for the vast majority of units.) In the case of a generator operating too slowly, the governor immediately requests more input power to compensate. Because this happens grid-wide, this gives the entire grid the ability to quickly respond to short-term fluctuations in operating frequency. Longer term frequency response is handled by dispatching individual units up and down to balance power and load.
In the event that a generator cannot adequately respond, it will trip (shut down) and disconnect from the grid entirely. This is a protective measure that keeps the generator from skipping cycles. (You can think of this as the electrical equivalent of gears skipping teeth, and can be hugely destructive to the equipment.)
The other cool thing about AC power is that you can send power based on the relative phase of a node on the grid. This means that a node at a lower AC voltage can send power to another node at a higher voltage by adjusting the relative phase. Mostly this is done by switching capacitor banks at various transmission nodes to adjust the reactive power component at that node.
Part of the problem is lower frequencies don't meet as much impedence in inductors (e.g., transformers), and another part is that interconnects can have control system stability problems keeping them synced.