DMZ and port forwarding are not compatible with each other, you chose either one of them
DMZ will forward EVERYTHING to the device you set up, port forwarding will only forward specific ports to specific devices.
Your router doesn't care about "recognizing your computer", it just wants to know which port has to be forwarded to another device.
For example, your router has a PUBLIC IPv4 address of A.B.C.D.
On your computer, with private IP 192.168.0.101, you have a web server running. Web servers listen to port 80 by default (or 443 if you're using https).
So what you want to do with port forwarding, is that someone on the internet can connect to your router on http://A.B.C.D and your router will forward the request to your web server on 192.168.0.101.
Hence you set up a port forwarding rule that has the following details:
Incoming port: 80
Forwarded port: 80
Forwarded IP: 192.168.0.101
If for example you have another computer with another web server (e.g. 192.168.0.102) that is also listening to port 80, you add another port forwarding rule to your router:
Incoming port: 81
Forwarded port: 80
Forwarded IP: 192.168.0.102
For the internet user, he should now enter the address http://A.B.C.D:81
When entering this url, your router will send the request to the web server on 192.168.0.102.
When entering http://A.B.C.D:80 (since 80 is the default port for web servers, you can omit the :80 from the url), the request will again be sent to 192.168.0.101.
Now if you create a DMZ and put server 192.168.0.101 in it, that means that ANY port will be forwarded to your DMZ. I highly recommend to use port forwarding instead of DMZ because it leaves your server completely vulnerable to the internet. EVERY single port will be forwarded to this server.
So you can ftp A.B.C.D and it will be forwarded to 192.168.0.101, you can ssh A.B.C.D and it will be forwarded to 192.168.0.101 and so on. From a security point of view, this is very dangerous and should not be encouraged.
Port forwarding give external access to a certain application (or port number) but you will keep vulnerabilities shielded off the internet.
You can forward multiple ports to a single machine as well. Some applications require the use of multiple ports. It's just not necessary to open ALL ports.
If you want to test if your port forwarding works, ideally you try to access your public IP on the forwarded port, e.g. http://A.B.C.D if you've forwarded port 80 to your server. When you do this, you shouldn't be doing this from your local computer (as traffic will stay local). You should either use a device that is connected to another internet source (e.g. your tablet or phone on mobile connection) or you can use a VPN connection so you can simulate that your origin is on an external network.
Note: You can use free VPN services, there are many, but be aware that they might snoop on your data (login/passwords/etc). Never trust free software/services, someone always needs to pay their bills and people usually do it with personal data instead of money.