I've tried all sorts of things for labels. Wooden sticks were a complete failure as no matter what I wrote on them with, the words eventually either wore off or went mouldy and became illegible, so I gave up on them.
Next up was the plastic ones, written on in indelible ink. The only trouble with that option is indelible does what it says on the tin. Yes, I know there are various methods out there to remove the ink, but to be honest that's too much faff!
The best solution I've found to plant labelling is to use the plastic ones and write in pencil - I like the 4" ones from B&M which come in packs of 50 and have a neat little pencil in the packet. It's a fairly soft pencil, probably a 2B or something like that, and I now write all my labels in pencil. It doesn't wash off, no matter how muddy and mucky the labels get, so I always know what I've planted.
Another tip I read recently was to make your own plant labels by cutting up those plastic cartons that milk comes in, and cutting the pieces into label-sized strips. Might try that one if I ever run out.