Now I don't understand wake on lan as well as I should, but from what I can gauge is that over the network (And potentially the internet) I can remotely turn my computer on, but that this is based on ethernet, and the network card is powered by the Ethernet connection, but is there any way that I can turn on my computer, in a Wake on lan like method, over wifi, on a laptop?
I could WOL my brothers desktop on wireless from pfSense and it worked fine but there were instances when it didn't work.
Could you go into detail?
Not much, the wireless solution was built into his motherboard and would accept WOL magic packets over WiFi, I'm not sure if it had an option in the BIOS relating to it or not like Ethernet but it did work.
You would have to have a wireless card which supported WoL in the drivers. So if you don't see any options for it then it's not going to work on your hardware. If you want to do it over the internet then I'm pretty sure you have to have something on your network which you can tell to send the magic packet to wake the other device, ie. You can't send a wake on lan packet over the internet (as it's a broadcast message).
I have heard of it being achieved before, and I have relative confidence that it can be achieved
There's definitely a way to do it. I found a fee tutorials when I was looking for a way to have Alexa turn on the pc. Which is really just wake through lan/WiFi since you can't have Alexa power pc from cold. Try searching that question and maybe you'll find the tutorial you need.