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Modem -> Router -> WiFi (option). The modem can be built into a Router or separate. WiFi can be built into a Router or separate, fed by ethernet.
That applies to Fibre, coax cable, DSL (copper phone lines) and even a 2 way satellite link.
Satellite isn't broadband. Fibre, coax cable, DSL (copper phone lines) are all forms of Broadband. Sometimes the Modem might use a Fixed Wireless link, this is rare Broadband.
The Mobile Cells are NEVER EVER Broadband, no matter how fast as it's not "always on", that's CDMA-1, GSM, EDGE, 3G, HSPA, 4G and so called 5G (more about infrastructure).
It's possible to have a standalone Mobile Modem, that might have WiFi or USB or Ethernet. Or you can use some phones this way (Tethering). My phone can work as a WiFi point for laptop or Kindle etc, but that uses its Mobile Cellular data, and then it's Modem and expensive. Not Broadband.
Ethernet is wired networking in your house. WiFi is shared radio connection in your house, WiFi has to connect to a Modem or Router + Modem to connect to the Internet.
You CAN have Ethernet and even WiFi connecting your computers, consoles, TVs, Kindle, printer etc and NO internet (Mobile, Broadband and Satellite are the three methods to connect to the Internet, which means "Interconnected Networks" and it existed many years before Websites, which are a resource on the Internet.
I might occasionally use WiFi on an eReader (I have not just Kindle) for a website or Wikipedia to look up a word or phrase I don't understand. I NEVER use it to send documents to my Kindle. I never use WiFi on the paperwhite or 3G on the DXG kindle for buying Amazon books. I always "download to PC" and transfer via USB. Amazon may think I'm using Windows, but actually the last two years I exclusively use Linux Mint.
The only difference in naming is that Europe we talk about Mobiles and Mobile Network and USA calls it a Cellular network and a Cell phone. Cellular is the technique used to implement a Mobile Network, so both are correct.
A Modem converts signals in the "outside world" to Ethernet. A Router allows more than one thing to share the Modem. (those can be separate boxes or in the same box). Networking via the household wiring is a marketing "lie", it's mains powered Modems with Ethernet and/or WiFi. They are really spread spectrum or OFDM transmitters. They'll work with no wiring up to about 30ft / 10m, the wiring supplies power and acts as an aerial to extend range.
The Kindle only needs re-registered if the account on it was deleted, or it is deleted on your Amazon web account. That needs WiFi, or possibly the Amazon Kindle App for Windows(Not sure?).
Once registered it never needs WiFi again, unless the account on Amazon OR the Kindle is deleted.
Also never report serial number of a Kindle to Amazon as mislaid, lost or stolen or it will NEVER register. Amazon blacklists and seems to refuse to re-enable. Then you can't create collections on it and only load via USB and any Amazon titles with DRM might not work.
The majority of my titles are either out of copyright from Project Gutenberg or my own writing. I do buy from Amazon and Smashwords. Via USB.
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