A wireless access point is a device that allows wireless devices to connect to a wired network using Wi-Fi, Bluetooth or related standards. It is basically just a source for wireless Internet connections.
The pure definition for an access point is distinct from that of a router : A router will also give such services as NAT routing, DHCP client/server, PPPoE client, etc. and will therefore create a network that consists of a unified IP addresses segment. That said, I do not believe that in access point mode your router will not provide these services.
A repeater is just a very normal client which, at the same time, can also be an access point, independent of the SSID and type of encryption used. A Universal Wireless Repeater is a device that you can place anywhere and it will wirelessly repeat the strongest signal, onto another wireless network (with or without security). As you have found, it implies the existence of two networks.
However, there is a bit of confusion in the terminology, as it is also normal to say that a wireless repeater extends the range of the local wireless network. These have also been called wireless expanders or range extenders.
is a forwarding technique used in packet-switched computer networks. Unlike routing, bridging makes no assumptions about where in a network a particular address is located. It is is only used in local area networks.
A bridge uses a dynamically-built forwarding database to send frames across network segments. It is basically used to virtually unify multiple network segments, by forwarding messages from one network to another with appropriate IP address translations.
In conclusion: It seems that the the people who wrote the manual and technical specs for your router should also have had a better education as regarding these terms.
I have looked at the manual of the Edimax. This is not a very recent model, but there is no reason for it not to function correctly. The manual says :
"Universal Repeater Mode" provides the function to act as AP client
and AP at the same time. It can use AP client function to connect to a
Root AP and use AP function to service all wireless stations within
its coverage. All the stations within the coverage of this access
point can be bridged to the Root AP. “Universal Repeater Mode” is very
convenient to extend the coverage of your wireless network.
It seems to me that "Universal Repeater Mode" is meant to bridge between two wireless networks, and therefore looks like the mode you want use, if you want it to connect to the router by wireless. However, I am still a bit worried since the image in the manual does not contain enough info to connect to the router, but that may be specified on another screen.
If "Universal Repeater Mode" does not work, the next to try is "AP Bridge Mode".
The text in the manual is very sparse and hard to understand. You will simply have to experiment with "Universal Repeater Mode" and "AP Bridge Mode" to see what exactly they do. It is quite possible that the shortness of the text comes from the ignorance of the documentation writer, and hence the exact terms he used should not be taken too seriously.