Should I emphasize my existing PhD in my master's application? Should the fact that I already have a PhD in another (useful) subject be downplayed on the personal statement? I'm concerned that it would make me look unfocused, but I also think I may want to emphasize it because it demonstrates prior success.

In case it's relevant, my PhD is in physics, and the Masters is in Computer Game Engineering. I am applying to a one year, full time taught Masters at a UK institution.
A person qualified in another area (such as a Phd in Physics) who wanted to learn a new topic (like computer game development) would likely be welcomed on a full-time taught masters course. Full-time taught masters courses are often designed for those who want a change in direction and have the motivation to focus for the year to achieve that goal.

There are people who want to be an astronaut, for example, who go back and learn new skills and knowledge and collect several degrees in the process.

You do have to have an interregnum in employment, as a full-time course is exactly that. A lot of daily, weekly and monthly time needs to be devoted to it to learn all that material that others will have picked up in a three year Bachelor's degree.

I teach on such a course and we have many "career change" applicants who are well qualified in other fields. It is quite normal.