This makes perfect sense to me the way you wrote this out. Why doesn't the diagram I was given with the kit have the woring this way? Maybe it does and I am not seeing it....
Supposing that you are using schematic S14746.R18:
1) A transformer with a 24VAC secondary --> see component "T1", location B4, wires 501 and 502
2) A fuse in one leg of that 24VAC output supply --> see fuse "F5", location D3, wire 501 in, wire 503 out
3) An emergency stop coil circuit that includes only OUT1 and the Emergency Stop pushbutton, as follows:
a) From the 24VAC L1 fuse to OUT1 COM --> wire 503 RED, locations D3 and F1
b) From OUT1 to one side of one of the E-stop button NC contacts --> wire 509 RED, locations F1 and D5
c) From the other side of that NC contact to the contactor A1 terminal --> wire 536 GRN, locations D5 and B2
d) From the contactor A2 terminal back to the 24VAC L2 --> wire 502 BLU, locations B2 and B4.
When you talk of "the E-stop", you need to be clear when you are talking about the pushbutton, and when you are talking about the contactor.
Also, your diagram does not show two emergency stop pushbuttons. It shows the circuit for one button, in the control pendant or console. That one button is connected to circuits on the control panel via a 4-position plug. The plug connection has two sides: one side leads to the button in the pendant or console; the other side leads to various points on the control panel in the cabinet.
The emergency stop button has two circuits going through it, via two separate normally-closed contacts. That is why there are four wires through the plug. One circuit, with wires 509 and 536, carries 24VAC power to the emergency stop contactor coil. The other circuit, with wires 901 and 902, carries 24VDC to INP11 on the PLC.