Terminatin and Safety timer are for two different purposes.
IF the TTC pin is fixed at any voltage it does not charge up and down for the counter to count and can't time out, which disables the safety timer.
The safety timer is an extra safety feature that can be used to shut off the charger if, by design, the battery should have completed charge and it has not. Typically the charge time is 2 to 3 hours and the safety timer is set to 5 hours so something may be wrong after 5 hours.
Termination is disable if the TTC is held low.
Termination is slighly better on the cell because it allows the battery to relax and does not constantly stay a 4.2V. The higher the voltage away from 3.7V the faster the Lithium decays. This may be the difference in life of 6 months over three years if left at 4.2V for the whole time. Probably not a big deal on a device like a cell phone because it stays off the charger most of the time. Note that if termination is disabled the charge current will go to zero as the battery reaches the regulation voltage and thus is not really charging at this time.
I will have to check what the status pin does since this indication is sometime slightly differnet on differnet part but if you test it, it should work the same each time.
My guess is that if the battery and source are removed and both inserted for the first time that the STAT1 will be low (LED on). I would think STAT1 would stay on until termination, which is disable so it would not turn off. It could be wired so the termiantion flag is set and the STAT1 turns off but the termintion is not acted (charger stay on).