Not communicate, just see that it was sending a signal that it wasn't before.

There's publicly accessible databases of all known satellites. You look up in a part of the sky and see a signal, you can look up what's in that orbit. For most, you can then find public information. If it's a satellite that's supposed to have died, and you're seeing a signal, well - that's interesting.

Again, lost contact. The exciting bit was that they started hearing from it again.

NASA only has so much capacity to talk to different space missions. The Deep Space Network system is in high demand, and they won't waste time trying to reach out to dead systems.

For instance, say tomorrow that Spirit or Opportunity Mars rovers started back up and were trying to phone home. Nobody is listening for them. They're 'lost' missions.

If, however, someone noticed this extra signal coming from Mars then there would be a whole lot of super excited people. You'd see more 'Lost Mars rovers found again' headlines.

The "I'm not a satellite engineer, I don't even play one on TV" explanation is:

Most satellites get their power from solar panels. Depending on their orbit, they might be in earth's shadow for part of their orbit. During that time, they'll need to run off batteries. Batteries have a limited capacity, and their capacity degrades over time. You design the satellite to only have the amount of capacity you think you'll need, and a little extra margin.

Satellites have management computers, with various levels of self-monitoring/recovery. Normally if something goes wrong, the self-monitoring will be able to detect this, and either "reboot" automatically, or go into a recovery mode where the ground-team can send up commands to diagnose and reprogram it.

However something went really wrong with this satellite and that system wasn't working.

The significance of the eclipse is that this would be a longer time the satellite was in shadow, and so the batteries might run down to the point where the satellite switches off entirely. Then, when the satellite is back out of the earth's shadow, the satellite will switch back on when it gets power back.

This would, hopefully, make the onboard computers go into a recovery mode, allowing that diagnostics/reprogramming.

It's sort of an extreme version of "Have you tried turning it off and back on again?".

Solar Eclipse = no sun.
Solar Flare = more sun.

"Honey, have you seen the IMAGE Satellite?"
"No, Dear, where did you leave it last?"

Lost as in "lost contact", not like they dropped it down the back of the couch and couldn't find it.

Equipment fails, so it's fairly normal.

e: Also, as for the price -- well more than a million, try about $132 million. "Costs for IMAGE are estimated at US$132 million, including the spacecraft, instruments, launch vehicle, and ground operations."

Looks like there hasn't been much progress - https://en.wikipedia.org/wiki/IMAGE_(spacecraft)#Recovery_efforts