Although it doesn't look it, my pen's tip is damaged.

I went to a reseller to try out my pen on another Surface 4. I immediately could see a difference in the end of the pen. The plastic around the soft edge was missing.

Googling images of the pen tip revealed the following:

This explains how I got to where I am. Read more on the blog with that image.

In my case, here's what the tip looks like:

As you can see, the hard black plastic is missing from the outside (it has broken off completely). This seems to be a design flaw in the original tip.

Short-term fix

I used an exacto knife and trimmed off the soft tip of the pen, so that just about 1mm was showing past the hard black plastic. The pen is working well again. However, I'm guessing this problem will return with continued writing because the exterior plastic is too fragile (at least for my writing).

I ordered the Pen Tip Kit (about CAN$25 online) and will see how the other tips fare. Hopefully Microsoft will address this issue.