I found a 1920's Atwater Kent Type E-2 speaker with the round connector in the middle of the cone separated from the cone. I used a little Gorilla Tape to reattach the connector and it seems to hold fine. What I find strange is the tab that vibrates the cone on the driver seems to move the cone left and right as it's facing you, not forward and backward like a traditional speaker. It's also barely audible with a modern stereo but does sound o.k. aside from the very low volume. Is this normal?

With a modern stereo or any 8 ohm driver, you will have very low audio with an E2. The E2 is designed only for use in pre-1929 radios that used high impedance outputs (1000 to 2000 ohms). Even with connection to an ancient radio, the E2 will only put out a maximum of less than a watt of audio; that's actually a reasonable volume level for that type of speaker and the intended purpose. The E2 will also have a terrible audio response compared to most any speaker made after 1935.

Pete AI2V

The cone should move in and out, though there may not be enough travel to see it. The vibrating reed should drive the wire along its axis.