You’ve basically confused yourself with a little bit of knowledge. You need to dive in deeper to get it cleared up again.
The modem doesn’t communicate to the router via it’s IP address. It communicates to whatever’s on the other end of the cable, and expects that whatever to sort things out. The modem doesn’t know from IP addresses. It’s not nearly that sophisticated.
Your router has two sides. One is the WAN side, where the modem is, and where the internet is. The other is the LAN side, where your computer is. The procedures and rules for the two sides are completely different and one has nothing to do with the other. The router’s purpose is to act as a bridge between the two.
Your ISP gave you a list of settings which you must follow in order to connect to them. That list said nothing about a router, because there are many hundreds of different routers and they don’t propose to even try to support all of them (or any of them). You bring in a router, you’re on your own. Which is basically reasonable of them.
It should be obvious though, that when you hook a router to the modem instead of your computer, that now the router has to follow all the rules and settings they gave you for your computer. If they said you have to use DHCP and PPPoE, then the router has to do that to connect to them. But that’s all on the WAN side of things. You hook the modem to the WAN connection on the router, and set the WAN side to follow their rules.
Over on the LAN side, you can use your computer to communicate with the router after you hook the computer to one of the router’s LAN ports. The router has a tiny web server built in to it, so you can configure it. You access it with your browser, by typing in the router’s LAN IP address. That address, or at least the default value for it, is in your router’s manual, which you should read carefully. If the default value was changed, and you don’t know/remember what it was changed to, then you need to reset the router to the factory defaults in order to access it. This will wipe out all of your other settings, so you’ll need to redo them on both sides. Your manual will tell you how to reset the router.
When we talk about setting a static IP, we’re talking about the LAN side, because on the WAN side you don’t have any options if you want it to work – you have to do things the way your ISP told you to do them. On the LAN side it’s all up to you, and you can pretty much set your network up however you want.
When you’re doing this setup, you either get somebody to do it for you, which is admittedly a little expensive, or you learn to set it up yourself. If the latter, you will get into trouble most quickly by blindly following lists of instructions without knowing what they’re for, and what you’re doing. You can avoid that by taking the time to learn. It’s really not a difficult subject, in fact it’s kinda primitive. The whole internet is pretty primitive, a side effect of the way it was developed. Simple fear gets in most peoples’ way, but it’s far less complicated than many other things you’ve already mastered, like irregular verbs. So just dive in. In a few weeks you’ll be wondering what you were making such a fuss about.