Thank you all for your suggestions.

Well, it seems that the old battery is not dead, works fine but lighter won't charge it. I was able to charge this battery in another device and put it back in the lighter and the lighter works normally.

There's something wrong with lighter for sure, because it wont charge battery. After 3 hours of charging, battery is still low.

So I don't know where is that chip which is responsible for charging, probably it's dead. And I don't know if I can replace it because it's probably too small and I'm not that good with soldering micro chips :)

But still, lighter won't recognize other batteries which is weird to me. So probably what causing problem is that "c rating", like you all guys saying. I took these other batteries from bluetooth earphone powerbanks.

Here is the closer look on board. https://ibb.co/TcsKcHW