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#11
Assuming you just swapped descriptions of 3 and 4, yes, that's more or less correct.
You mean using the same cable and connecting it to one computer at a time? Seems like that thing you have is a modem. They are the cheapest of the bunch, and are an obvious choice as customer freebie for ISPs.
Your choices are to buy a wireless access point (that you forgot in the list, they simply connect wifi devices to a ethernet network, that may have access to interent or not depending on other devices) or a full blown router with wifi. The latter has more ethernet ports so you can connect multiple computers at the same time, wifi antennas for wifi devices (obviously), and usually the features most routers offer (and modems don't have) like security options, firewalls and whatever.
You can also buy the "3" from your list, and ditch the modem. Although that requires knowing precise data about how to connect it to your ISP correctly (as you need to feed the right data to the device's modem for it to work). Most ISP-given modems are set to run automatically, so you will have to ask your ISP how to set logins and passwords and specific settings you need, it's a bit of a hassle, but it's usually worth it.
Btw, with wifi ALWAYS have a good password and buy only the devices that use WPA or WPA2 encryption (laptops all do, not all wifi access points/routers do). WEP is a joke to crack, and WPS (the "push a button and the network is secure") is only marginally better.
Really, knowing a bit in the field of wireless network security isn't so hard and does pay off (as wifi network cracking is pretty common if compared to guys that connect illegally to a wired network, lol). Google around and you'll find plenty of sites that do explain the basics.
There is also another kind of box that you might be interested in, called network (or ethernet) switch. It is basically a dumb router, it has no settings and no added features (like say security or firewalls or whatever), and simply routes traffic through the ethernet cables. While a router does act as a network cop, this is more like a traffic light. Companies with decent networking needs will have truckloads of these things scattered around, but will have at least one device something doing the router job as well (either a router or a full blown server).