WLAN repeater or access point?

Both are correct, depending on the technology used.

Well, 2 APs have the disadvantage that you have to manage two of them. There is a CAT cable at that spot as well, but it was unnecessary. And why buy a PoE injector just for that?

Why manage 2x? For me

Well, 2 APs have the disadvantage that you have to manage two of them. There is a CAT cable at that spot as well, but it was unnecessary. And why buy a PoE injector just for that?

What do you have to manage on an access point? It just passes it through! Assign the password once and done. It's best to work with fixed IPs from the beginning.

Unfortunately, that is not true. Every classic repeater in your network massively reduces the data rate because it cannot send while receiving packets. So with a repeater, the packets are in the air twice as long, thus the data rate is halved. The interference with other networks is doubled.

Have I claimed otherwise?! Who cares whether half the packets get through or whatever. The questioner is concerned with the technology to be used, and that is clearly a LAN cable from the router to the "Wi-Fi amplification point." Whether the packets are halved does not help the questioner at all. For the LAN connection, also a power connection. That way you play it safe and can connect everything there. You could later also set up a NAS there or something. I solved it like this:

A NAS on the ceiling, I would like to see that
Basically, you can of course plan for power, but if you plan PoE from the start, you can safely leave out the additional socket.

In the basement is the star point where all network cables from all rooms meet. A router, for example a FritzBox 7360, is planned there. On the ground floor in the hallway under the ceiling about 30 cm, I plan an extra network connection + socket, where an access point is planned.

It makes sense for all the network cables to come together in the basement or utility room, correct (star wiring). At this point, a patch panel is installed, and the network cables are terminated on the panel (backside of the panel). A switch is then added, and short LAN cables connect the panel to the switch. The switch is also connected to the router (e.g., a FritzBox).

The FritzBox does not have to be located in the utility room. If you install double LAN sockets, you can, for example, place the FritzBox in the hallway, then the LAN cables run from the FritzBox to the utility room and back again. Advantage: better WLAN and DECT coverage.

For WLAN coverage, I would then, as suggested, plan an access point approximately in the middle of each floor. For this, I would have a LAN cable installed, possibly not connected to a socket but rather directly crimped with a plug. The access point is then connected directly. A switch supporting Power over Ethernet (PoE) is required. Possibly, two switches are used: a large one for the regular network and a small PoE switch for the WLAN (and the IP camera, etc.).

Switches that support PoE are quite expensive - I have a Cisco model here, but I wouldn't recommend it for a regular household. I'll probably switch soon anyway, because the device simply draws too much power for 7x24h (60-70 watts). There are injectors that can do it cheaper - well, then you have passive PoE and therefore no Gigabit LAN, but that shouldn't matter.

Crimping a plug onto the installation cable is possible, but it's not nice - especially if the cable needs to be bent. What's wrong with a proper socket?