When lip filler is injected into the lips, is some filler always injected into the lip border?

No.  The vermilion border should only be injected if needed.  It is customary when one is learning how to do injections that it is an approach that is taught.  I find that many injectors who are starting out feel like this needs to be the first to do.  This is a mistake that you have experienced yourself.  I would recommend that when you consult with an injector to ask them what they see in your lips and what they would do before considering them to do your injection (hopefully it matches your expections).

Filler doesn't have to be injected into the boarder. It is patient preference or desires that determines where the filler is placed. There is limited quantity of filler in a syringe and injecting the boarder will utilize a few tenths of the 1 cc when it could be used elsewhere. If smoker lines are an issue, then it is usually used for the boarder. Injecting in the boarder or philtrum will give a more " duck beak" look

Thank you for your question. There are eight different areas in the lips that can be injected. Those eight areas, like the vermillon border, do not have to be all injected. It depends of what the lips need for enhancement and ensuring the results are natural looking. I hope this help. Good luck,