I know you already wrote that you are able to configure your old AEBS with the "launcher app" but I am not familiar with it. If it is a modification of OS X as I suspect, be advised that there have been a number of users reporting that doing so eventually left their system with no operable version of AirPort Utility, leaving them in an untenable situation. Should that occur, you must first satisfy yourself that you will be able to recover from it. A future OS X update or upgrade potentially changes everything. I have reason to believe Apple no longer supports the original 802.11b/g base stations because their hardware cannot meet iCloud's robust security requirements. Given that iCloud is so inextricably intertwined with many of their products and services, it is reasonable to surmise Apple would prefer all those devices be removed from service.
With those caveats out of the way, yes, reset the AEBS by holding its reset button depressed while connecting it to power. A few seconds later all three LEDs will illuminate. Then, release the reset button. Use AirPort Utility to configure it, and when you do you will need to select the AEBS from the left column, then the Internet pane at the top, then Internet Connection. "Connect Using" must be set to "Ethernet". "Configure IPv4" should be set to "Using DHCP".
Next, select the AirPort pane from the top, and from the Wireless Mode dropdown menu, select "Create a wireless network". Provide the exact same wireless network name, security, and password as your existing network.
I no longer have any round AEBSs to test and that is all I can recall. There may be a step or two I neglected. If you run into a problem either describe it or post a screenshot, and I'll see if I can determine what's wrong. Usually, if something is misconfigured and you cannot connect to that AEBS, it's best to start over again with another reset.
