You definitely want an ink jet printer for labels. However, the type of ink in the printer makes a huge difference. Regular inks are water soluble, whereas pigment inks are not. Here is a link to an article by Gloria Hansen that explains it:
https://www.gloriahansen.com/articles.php?p=1&ID=14
The ink we use in our Brother printers is pigment ink, so should last. I found this out by looking it up on Amazon (which is where I buy our replacements).