What is the difference between a name and a DNS name?

In the DNS configuration of my router, I have this table where I can change the dns name of the devices: What's the difference between name and DNS name? Aren't both supposed to be the strings that you can use in replacement of their IP to refer to them? If a DNS name is just another way to refer to the IP, How come I can't use, say, 5 different names to refer to the same machine? If I had a server hosting 5 different domains, Shouldn't I be able to put all of them in this DNS table? But I only have one textbox next to each device...

What's the difference between name and DNS name?

It's one of those manufacturer-specific things that, without knowing the router's model, I can only try to guess about. (Though even if I did know the router's model and downloaded the manual for it, 2/3rds of the time it wouldn't be in the manual and I'd still have to guess.)

But in this case, the 1st "name" column is the name that was provided by the device (in the DHCP lease request), so I would strongly guess that the 2nd "DNS name" field is meant to allow you to specify an alternative name, e.g. so that you could use the name "printer" instead of "HP54C850".

(Of course, computers can be renamed within their OS, e.g. in Windows Settings, but most appliances cannot – their "advertised" hostname is either just the model name, or something based on the serial number and/or random-looking.)

However, it's unclear from this screenshot whether the router automatically adds the devices' "DHCP" names to DNS and allows you to specify alternatives/aliases if wanted, or whether it doesn't add any names unless you manually specify something, but it'll probably be something along those lines.

If a DNS name is just another way to refer to the IP, How come I can't use, say, 5 different names to refer to the same machine?

You can. In a "fully featured" DNS server.

Your router is only barely a DNS server. It does the absolute bare minimum it can get away with, while still providing the one feature that users have come to expect (collecting the hostnames from hosts' DHCP lease requests and making them resolvable via local DNS).

Note how it also doesn't let you edit the IP addresses, or add subdomains, or add any other DNS record types except for 'A' (for "IPv4 address") – it's made with a specific feature in mind, and it's not made to be a generic all-purpose DNS server, just like the router itself isn't made to be a generic all-purpose router.