Is my wire diagram correct?

Pretty sure they are mechanical switches with added circuitry
that means you are intended to wire them like PNP proximity
switches

You need to figure out how those switches work. The openbuilds page and their video both don't say anything about how they function, also didn't see an LED in the video. I would avoid buying anything in the future that doesn't have a wiring diagram or more specific information. To test the switches hook them up to the 5v psu and then use a meter to check and the signal wire vs the ground wire. I expect the switches go to 5v when activated. If thats the case then you need to wire things the following way.

5v PSU- to TB3 PIN12 Input Common

5v PSU+ to Limit switch V+
Limit Switch Gnd to 5v PSU-
Limit Switch Sig to TB3 PIN2 (second switch to TB3 PIN3)

Using a E-stop that has two NC contacts will work as follows for the E-stop input:
5v PSU+ to E stop to TB3 PIN1

E-stop for relay control for the router using the 2nd contact on the e-stop switch:
5v PSU+ to TB3 PIN14
TB3 PIN 13 to Estop to relay+
Relay- to 5v PSU-

One of the nice features of the 7i96 using 5v for your field wiring is that you will have a number of +5v (protected) and commons for your wiring needs and won't need a terminal block or similar for wiring solutions.

So anytime I mentioned +5v PSU you can actually use any of the following pins:
TB1 PIN6, 12, 18, 24
TB2 PIN6, 12, 20

In place of 5v PSU- you can use:
TB1 PIN1, 7, 13, 19
TB2 PIN1, 9, 15, 23, 24