Vinegar or salt will not set modern dyes. Dyes used to be organic and vinegar and salt were mordants for some dyes. Modern dyes are chemical and vinegar and salt don't affect them in any way.
Your white towel acted like a color catcher. Some types of fabrics will pick up excess dye in the wash water and some won't. It's not that the color is bleeding either. The fabric just has excess dye and the excess comes off the first time you wash it.