Which plastic bottles are are safe?

Unfortunately, that answer is complete nonsense.

The plastic used to make drinks bottles is polyethylene terephthalate, or PET. It has a melting point of around 250 degrees C - a hot drink is going to be less than 100 degrees C - at that sort of temperature PET doesn't even soften, let alone melt.

Secondly, even if you did melt PET, it would not be miscible with water (and all drinks are water based). The PET would need to be at least 250 degrees C - try to mix that with a water based drink and all you would get is a very rapid "Whoosh" of steam and the drink would evaporate.

It is, therefore, quite impossible to drink water "with plastic in it" - it is not physically possible to dissolve synthetic plastics in water.

This sort of nonsense probably comes as a result of scaremongering over the use of PET bottles, where the usual comments made are that they "leech things into the drink, like aldehydes". The reality is that, yes, PET can decompose to form aldehydes but only at very high temperatures (above the melting point), and in normal use the bottle would never get this hot. So this is a product of thermal breakdown found due to improper handling. There is also a lot of talk about the element antimony being "leeched" into water by PET. Again, yes, antimoney trioxide is used as a catalyst in the manufacture of PET, and a minute amount (parts per million level) of antimony containing species can find their way onto the surface of the plastic. However, in normal and correct use, this is removed by washing before the bottles are filled. So once again, if there is an antimony contamination issue it is purely down to incorrect handling.