PoE is Power over Ethernet normally.
Basically normally an ISP gives you a gateway which allows you to connect to the internet. This is a router and switch in one. The mac address of the gateway is allowed access to the network. So if you connect a new router it won't be able to connect.
With my ISP I can set my gateway to modem mode and connect my own router. So long as I have my own router connected to the right port on the modem I will get an IP address assigned and have access to the internet. Sometimes, if I want to change the router, then I must power cycle the modem. The modem acts like another router and has DHCP.
With cables, he is using a normal cat5 straight cable? Normally it does not matter, good network devices can detect and work around some cable issues. Like the wrong cable type, for example crossover needed but straight through cable used. Router to router connections require a crossover cable. My modem and router can work with straight through cables. So can my Cisco routers when they are connected together.
He states, "the router doesn't seem to be able to detect the cord" There are no lights which implies there can be no hope in the world. There are two reasons for this to happen, 1. there is no physical connection on the internet port on the router and/or 2. The router is not powered on or the device at the other end is not powered on. Sometimes it can take a network device a long time to boot. With my modem its only at the end of the boot cycle that the physical ports start working. This takes a few mins.
So when he connects the laptop to the modem then the internet works? This connection would require a crossover normally. You have a Router connected to a workstation.
We know at least one router is powered on because of the statement "the router keeps saying that it cannot detect an internet signal." Is the laptop connected via wifi to the router?
Now if the laptop can connect to the internet via the port on the modem then the internet port on the router should be able to get an IP address and default gateway.
The ports on the back can be seen here,
https://www.linksys.com/us/p/P-EA7300/
You need to troubleshoot the physical setup at this point. You may need to contract your ISP for help.