Actually seeing the pattern is helpful.
I've not encountered any instructions for kb or k2b where you don't move the st off the left needle unless instructed otherwise which can be used as an increase. Looking at the photo of the cowl and the instructions, I say that you work your k2b and move the st off as if you'd worked a normal k. This is what produces those elongated stitches. Try it and see how it works. You could do a swatch of stockinette for experimenting on before you go round your cowl multiple times.
ETA: Just having had a closer look at the pattern, you might prefer to use purl sts between the k2b as in the pattern for a test swatch. A similar effect can be had by slipping a k st every other round.