Yes, the first browser was WorldWideWeb, written by TBL himself. Then, there was a small ecosystem of other browsers before Mosaic came into the picture. Mosaic was the first browser with a "must-have feature" which allowed it to become very popular compared to the others. This must-have feature was inline images, using the <img> tag. Before this, if you wanted to include an image to a page, you linked to the file with an <a> tag. After Mosaic became popular, the team who developed it started Netscape, which was unofficially called Mozilla, i.e. Mosaic K*lla. This was sort of the "zeroth" browser war. Netscape's must-have feature at the time was frames using the <frameset> and <frame> tag. After that, came IE and the "first" browser war, between IE and Netscape...

(I wasn't there, but I've done a lot of research into this.)