A fresh alkaline 9V battery can easily fry a lot of smaller LEDs. It can deliver a couple of amperes into a short circuit. A zinc-carbon 9V battery has a considerably higher internal resistance (hence their unsuitability for certain applications) so the LED may well survive (although possibly with damage).
Not-so-fresh cells have higher internal resistance, so even if they measure a similar open-circuit voltage, that does not provide a good indication of what they will do when connected across the hapless LED.
Most Li-ion cells can deliver quite a bit of current (low internal resistance) so it's not surprising the LED did not survive.
At the other end of the spectrum, a CR2025 or CR2032 cell has enough internal resistance (and a lower ~3V voltage) so that most LEDs will survive without damage, so one sometimes sees vendors at electronics markets demonstrating LEDs by connecting them directly across such a cell.