The bit that Dremel markets as 'tile and masonry' which is a sharply-knurled carbide rod is ideal for drilling glass. The 'glass' bits are more for engraving.
You must fill the bottle with water and sink it up to the level of the hole in another bucket of water as you drill, to do 3 things: Cool the glass (cracking), cool the bit (dulling), and damping the vibrations (cracking). Use a plug-in type dremel tool as the battery ones don't get up enough speed to do the job fast.