How to knit a thumb on a (fake) mitten?
I tried the instructions from your pattern as written, and it gave me a single hole with a string running across it. That string would be a problem in a real mitten, but it might be okay for a decorative one. Presumably the pattern will have you pick up stitches around the edge of the hole and knit a thumb. I have no idea how you're getting 3 holes. Please add a photo of that, and maybe we can give you some more customized troubleshooting.
A normal way to make a thumb opening is to bind off stitches on one round. On the next round, cast on an equal number of stitches. So for your pattern, it would go like this:

Thumb round 1: Bind off 3 sts, knit to end of round
Thumb round 2: Cast on 3 sts, knit to end of round

After the main part of the mitten is done, you would pick up 6 or 7 sts around the edge of the thumb hole, knit several rounds to make a tube, then decrease with one round of all k2tog, then break yarn and draw it through the hole.