So the thing my laptop is connected to right now, over a wireless connection, we call a "router" where I come from. Almost everyone has one at home here in Switzerland and a typical household consists of roughly 5 people or less. So about 5 people are connected to this router, which absolutely disagrees with the above statement.
There is no lower limit as to how many devices can or need be connected to a router. Its purpose is just to connect separate networks, and there's nothing unusual about having e.g. a 2-device network.
So corporate networks do have routers connecting thousands of devices, but your home LAN indeed has a small router as well, and if you've ever used the "hotspot" or "USB tethering" feature in your phone – it turned the phone into a router too.
Is it simply wrong to call that thing that provides WLAN a "router" or what am I misunderstanding?
These used to be called "wireless routers", because they had "wireless" (Wi-Fi) and "router" all in a single box. And the point I want to make is that the thing that provides WLAN was the "wireless" part. Not the "router" part.
Words do shift over time, so for clarity: when the book says "router", it's referring to a device which deals with IP packets and IP routing tables – it isn't referring to anything Wi-Fi related.
As to whether it's wrong to call that thing that provides WLAN a "router" or not – it depends. At home, most of the time, it's indeed the same device that acts as an IP router and provides Wi-Fi. But for example in corporate or university buildings, you will see many Wi-Fi devices which don't perform any IP routing whatsoever; they are just Wi-Fi "access points" but not routers. (And vice versa, all those access points are connected to a router which has no Wi-Fi capabilities, it is just a wired router.)
(If your "main" router can't provide good Wi-Fi coverage for the entire house, it is common to just chain a second wireless router off it, creating a whole 2nd LAN that way. However, many posts here will recommend improving that setup by disabling the actual routing features of that 2nd router, turning it into nothing more but a Wi-Fi access point/bridge (giving you advantages such as seamless roaming from one to another). In this case, the device both is and isn't a router: it was built to be a router, but isn't acting like one anymore.)
So it is not possible to be connected to the internet over a switch? You do need a router, right?
Technically you don't, but in practice you kind of do.
You don't need a router, technically speaking. There will be routers somewhere upstream – the Internet is literally made by interconnecting networks using routers – but not necessarily at your home.
However, if it's a home connection, then often your ISP will only be issuing one IP address over your line. When that's the case, you can directly connect one computer to the ISP – or the other computer – but not both at the same time.
Home routers bring another function – NAT – to deal with that. They let your home network use "private" addressing, which the ISP doesn't care about and is unaware of, and when the packet goes through the router they translate it to your "public" (ISP-issued) address.
This NAT is one major reason why a router is usually necessary, due to sharing that one ISP-issued address, though it does bring other benefits as well – creating a separate network lets you can freely decide on your own IP address assignment and everything; the router also has useful things such as a DNS cache; and so on.
In a way, the same goes for big corporate networks. They might not technically need a router between their network and the ISP, but they will have one to specifically establish the separation between their network and their ISP.
And is the modem considered as another node in the network?
There's no simple answer to that either...
In theory, a modem's job would be to only convert between two kinds of signals, not really caring about IP addresses and not showing up as a "node" at IP level. In practice, that doesn't always happen.
Many modems nowadays are in fact combined with a router (see previous section). It's fairly rare to be issued two separate devices; it's cheaper for the ISP if the modem can just do routing on its own. Then the modem does actually show up in IP traceroute and generally behaves just like you'd expect a router to behave.
It also depends on what kind of modem it is, i.e. what type of network technology (dial-up? ADSL? ADSL with PPPoA? VDSL? fiber Ethernet? GPON? 4G/LTE?) it's dealing with.
Some WAN connection types are built so that they can be directly bridged to Ethernet, allowing the modem to act as a "pure" modem – for example, with some ADSL providers or with GPON you can directly attach to the ISP's network, with the modem being just a bridge.
But on the other hand, 4G/LTE does not allow this; the 4G modem always needs to care about IP, so if its purpose is to connect an external device then it has to be a router.
we actually have three "routers" at home (devices that provide wlan), are they all combined modem/routers?
No, most likely they aren't modems, in the usual sense. And if they are, then they won't be using the modem functionality in the middle of your LAN.
But in a network of thousands of devices, we could theoretically connect them all to one router like in my network at home without the need of additional routers? It just wouldn't work very well, right?
You could connect a lot of devices using just plain switches, yes. Though there will be scaling issues at some point. Just a few examples:
IP routing is hierarchical and prefix-based, with all addresses within a network sharing the same prefix – so a single route can take care of any amount of devices. For example, if you have 10 subnets, then the routers need at most 10 routes to know where everything goes (possibly even fewer as routes can be aggregated) – even if each subnet has thousands of devices.
Ethernet switches don't have that. They only work with MAC addresses (associating them with switch ports), and MAC addresses within a LAN have no relationship to each other, so the switch has to remember every one of them individually. Often a switch will have the capacity to remember 4k or 8k MAC addresses, so things will start going wrong if it has to deal with 10k at once.
Most PCs and other devices tend to send out a small bit of broadcast packets to the local network. (Some of those are critical – such as DHCP requests or ARP queries – and some of those are not. For example, Chrome will occassionally scan for nearby Chromecasts.)
Broadcast packets do not go through a router, but they will go through all the switches. With thousands of devices connected only through switches, that adds up to a lot of packets that have to be delivered to every device, whether it wants them or not.
