I'm always saying, "I forgot to charge my phone!" I always forget to charge my phone. So it dies. Ah! Poor thing! You can't run out of battery ... but you can run out of battery power. You, personally, can run out of energy! (you're tired). You can run out of milk for your coffee. If you don't have any batteries at home you could say, "I've run out of batteries!" Then you'd have to go to the shop and buy some batteries (for your torch/clock/whatever!) You could say, "My battery's dead!" or "My battery's died!" or "My battery's out of juice!" I'm sure others will think of all the different ways we say our phone's not working because.....

Yes, it's fine! You can also say: My phone died. Or my phone ran out of battery.

That's right.