Elizabeth, Yes, you continue the pattern on the purl side. In some ways it's easier on the purl side because the strands are on that side (even though the pattern is easier to see on the knit side). And yes, fair isle in the round seems simpler to me.
Jan, wait, did you not attach the video?
My beginner fair isle pattern was something I made up. It's done with two colors and is kind of a fair isle stripe pattern. You can use any two colors. I had one solid color for the main and my second color was white. I knit in stockinette four rows of the main solid color. For the next four rows, I knit one in the main color and one in the white. On the other three rows, I alternated the colors. Then I did another four rows in the solid color. 0 is the main color and x is the white. It's something like this: