Thanks. I tried both options for a few days, and I'll settle for keeping the same SSID. I found
another page on the DLink support site which states: "D-Link suggests creating a new wireless network name and password for the extended signal because it will allow you to easily distinguish between your "primary" signal and your extended signal".
Now, indeed, with a different name, you can clearly ensure that you're on the extended network, and thus have the best signal. The problem is that devices will not automatically switch to it if the "primary" network is still in range, even if the signal is weak.
With the same SSID, it's much more transparent, and you don't need to worry about it. When connecting a device, it selects the strongest source. If moving around, not all my devices switch directly to the strongest (my laptop does, my tablet doesn't), but switching WiFi off and back on forces it quite easily.
I just wanted to share that experience.
Cheers!