There are a number of different options with the way that an E-Stop can function, and also depends on the type of machine and if it is an industrial operation or home. The safety E-Stop systems in industrial settings use double switches in a serial configuration, or two separate loops along with a monitoring loop. If any switch fails or power loss it goes open, the purpose to prevent any potential of a system restart with dual redundancy. In addition often an emergency stop button's purpose is to shut down the machine as fast as possible to prevent injury. This often can be a destructive mechanical braking system. I always use a dual circuit E-Stop in my designs, which does three separate actions: deactivates power to the system run relays which are used to activate the VFD run inputs, cuts power to the latched power relay which also stops the the possibility of the VFD receiving any run command and also activates a emergency fast stop to the VFD. In some cases a VFD reset may be required depending on the programming. There are also various other interlocks which always require the machine to be in the stop mode when there is a fault and a reset is performed. This is similar to what you see with factory lathes and mills, that have relays/contactors and/or VFD's. Additionally there are other fault triggers such as the belt cover switch, chip guard or foot brake that once activated require a reset to prevent a restart.
The GS20 looks like a nice reasonable costing replacement for the older GS1/2 VFD's with a lot more features, and nice to see that it has the circuitry for an external braking resistor. The Teco L510 is an inexpensive VFD, which works ok, but does not support an external braking resistor. It is also quite limited as to the number of inputs and programming. The Teco E510 is more comparative to the GS20, the price of the E510 has gone up significantly this last year, so the GS20 is a good choice and Automation Direct has great support.