Download on one Xbox on the house same name while playing on another?

If you have your console set as your Home console, the games should automatically download there. Otherwise, you can't be signed into both consoles at the same time to download it like that.

The most interesting information that we'd need to know is, if the wall ethernet is a direct public connection to the internet (e.g. an University Dormitory, where each room has it's own public IP), or if you are living in an apartment complex, where the
whole complex is connected to one public internet line, and the different apartments are getting a "private" IP address from the service provider at this apartment complex.

In Scenario 3, what is the router being plugged into to get internet that it then distributes via WiFi? Is that a travel router where the router gets Internet from a cellphone based service (e.g. 3G/4G/LTE)?

Since in the first two scenarios, both consoles are eventually connected to the wall ethernet in one way or another, they are both connecting to the same network that provides internet to the wall ethernet plugs. So it might be, that the ISP running the
"wall-ethernet-internet" is not correctly
Hairpinning.

I assume in the Scenario 3, someone uses a router that is not connected to the wall internet, and gets the internet connection from a completely different internet connection (I'd assume cellphone based, or if there is a public WiFi
at the apartment complex, it's a separate IP network from the "wall-internet").

So if you could provide us more information about the infrastructure behind the wall ethernet connection, I might try to help a bit more.

One of the consoles is connected directly into the wall Ethernet port. It gets internet and everything but it doesn't let us play together. Is there an alternative to spending more money on another router?

I'd recommend checking on the NAT status in network settings on both consoles as you will want to ensure both consoles show an Open NAT and not Strict etc.

Open NAT if you don't know is purely to ensure you both don't experience matchmaking issues/party chat problems etc. Also, it will be worth hard rebooting both consoles and doing a reboot of your router to ensure all caches are cleared on each device. Then
boot one back up, check NAT, leave it switched on and boot the 2nd console and check the NAT.

Easiest way to attempt getting open on both consoles is logon to the admin page for your router at home, take out any previous firewall rules you created specifically to get the 1st console in Open NAT (if you had to)/disable the DMZ (again if you were previously
using this method for an Open NAT) and just enable UPnP on the router, if it supports it. That is the easiest method of getting both consoles working fine on the same network, however, some routers UPnP implementation is a bit naff and they still struggle
to give both consoles Open NAT. At which point it might be best to look at the age or model of your current router and compare with newer models.

Sorry if I've misunderstood something but both consoles have to eventually reach a router. Even if its through the wall it has to go back to a router at some point to then make it out to the incoming line from your ISP.