How to paint a textured brick fireplace

Tyler asks,

My house is in Evansville Indiana and was built in 1908. I want to paint our brick fireplace white. I know painting means it will be permanent which is fine with me. However, I saw that there are white brick stains and wanted to know if a white stain would be better than regular paint?

Also if paint is the better option. what type of paint and sheen should I use for this? I planned on using a paint sprayer for this since we’re redoing the whole room so I’m not worried about over spray. I planned on cleaning it with a TSP solution before I paint/stain it. If I painted I was going to prime with a gray primer.  Thanks for your help!