Which internet Wi-Fi mesh system?

That has absolutely nothing to do with each other and you definitely don’t need mesh here. Just google mesh.

What is mesh then in your eyes?

What is mesh then in your eyes?

Networking of WLAN access points via a second/third radio channel (wireless backhaul). A common SSID and good coverage can also be achieved through wired access points; for that, you don't need mesh.

A mesh router system is, in my eyes, as @ explained, a simple solution

For example, I have 3 routers/repeaters like Netgear Orbi, Amazon Eero; there are many 3-piece mesh sets. (they are mostly advertised as WLAN mesh sets or systems)

Each device would also be connected by me via LAN, and no redistribution via WLAN

Question about the access points: are they always connected via POE?

Question about the access points: are they always connected via POE?

Most of them.

I have for example 3 routers/repeaters like Netgear Orbi, Amazon Eero, there are many 3-piece mesh sets. (mostly advertised as WLAN mesh set or system)

Orbi works and is an easy way, I have it running at my parents' house. However, they need a normal power outlet, no PoE. Also, you have to want the look of those things. My solution wouldn't be that in a new building; there I would definitely rely on access points on the ceiling.

ok thanks so both ways work well?

Where would you place the Access Points?

I somehow can't imagine having an Access Point on the ceiling in the living room.

One Access Point on each floor? what range do they usually have?

Do the Access Points light up?

So you have little knowledge of the subject. Mesh is a kind of network setup. It is slower and less stable. You simply need at least one WiFi access point per floor. More rather if you want good coverage in fast 5 GHz WiFi. You let each access point broadcast the same WiFi. In other words, the same name and password everywhere. Your client (phone, tablet…) then connects to the strongest signal. When you walk through the house, it jumps from access point to access point. There are also access points for the wall. But the ceiling is better as there are no obstacles there. They definitely don’t belong right behind cabinets!!! That takes away your fast 5 GHz WiFi and you get stuck with slow 2.4 GHz WiFi… What do you mean you have enough LAN ports installed? Everything that has a LAN port and is stationary should go into LAN. Only mobile devices into WiFi. There are already some posts on this topic.