Congratulations first!
To me, if the ultimate goal is doing data science, I prefer to get a PHD first. From what I got from data scientist recruiters, people with a master in DS basically work under the lead of PHD researchers and masters usually take up the validation or the programming role. The description on Stanford's website gave me an impression that the program provides a good range of career paths (CS, DS, stats, etc.) and students can also choose to pursue a PHD after the program.
If you are worried about the future job market, I think PHD is far less risky than a master unless you are a good programmer (yeah, programming skill is important for masters) and the skills you learn from PHD are transferable (IT industry and even hedge funds!). UCLA is also a great university in terms of statistics. But, as you mention, getting a PHD takes a long time, so it is important for you to know if the supervisor and his\her projects are of your interests.
On the other hand, if you want to have a career path other than quantitative fields, Stanford is possibly better because it provides you networks to do something like starting your own business.