You can do either, but the normal thing to do (e.g. for companies that have many wireless access points) is to use the same network name for all of them. This way you don’t have to choose the right network for the part of the house you’re in - the computer will just do it. If there is any overlap between the signals at all, it is important that they be on different channels (ideally non-overlapping channels; the easiest way to do this is to use 1, 6, and 11, but you can choose other channels as long as they’re far enough apart). If one of the access points is a router, the other definitely needs to be in bridge mode. But definitely use it as a wired access point. The wireless network extension can work, but it is never as good as a second wired access point. It’s slower, and can be flaky. It’s the last resort for when you don’t have a wired connection. Since you do have a wired connection, there’s no reason to use it.
One thing to know is that large enterprise wifi networks have “controllers” that manage the access points, as well as other tricks that help clients transition seamlessly between APs. With two standalone APs in your house, you won’t have this, and depending on the devices you’re using, the handoff might not be as good as you’d like. For example, if you start out near one AP and then walk to the other, many devices will hang on to the first one even as the signal gets very weak, rather than switching to the second one when it becomes better. If you don’t roam between the areas much, this isn’t really an issue, and even if you do, it might work well enough for your needs. If it really bothers you, you CAN set them up with different network names, and then just manually choose the one that’s better for where you are. If they both have the same name, it can be hard to force your device to associate with a specific AP (and its ideas about the best to use may not agree with yours). Though again if you’re really in a situation where you have no signal from one AP in the other location, it’s not really an issue.
Another thought… If you have ethernet cabled from your router to your second AP location, does this mean you have ethernet throughout the house? Depending on the house layout, it might make sense to turn off the wireless on your router (use it solely as a router, not a wireless access point), and put a high quality wireless access point at some central location in the house, with a wired connection to the router. With that, you might be able to cover the whole house with a single AP. My house is on the order of 4000 square feet across two floors, and I cover my whole house with one AP. But it’s an enterprise-grade (read: expensive) AP; I tried it with the one my cable company gave me, and there were several parts of the house where I effectively had no signal. The nice thing about a single AP is that it avoids the channel overlap and roaming problems entirely (aside from any channel conflicts with nearby neighbors).