I am trying to setup a private wireless network in the form where my laptop is connected to Wi-Fi with internet access:
Wi-Fi (with internet) > Laptop > (Ethernet) > Wireless access point > other devices
to enable the other devices to connect to the internet through the laptop’s Ethernet-connected access point.
The reason for this is that the original wireless network is WPA2-Enterprise encrypted and I have devices such as my Xbox and Apple TV that do not support this. (I can’t simply change the wireless encryption as it is my university supplied Wi-Fi.)
Do I need to bridge the laptop’s Wi-Fi and Ethernet connection … or enable internet sharing?
For your proposed routing to work you need to enable Internet sharing. A more elegant (but more expensive) solution might be to get another Access point type device which you can use as a AP Client (or something which can be made to do this with DD-Wrt or equivalent), and then take the ethernet out of that and into your AP.