How do you differentiate STAT/ML/DS in SOP?
wouldn't it be weird to say "Through my research experience in DS/ML, I developed an interest in STAT, so I'm applying to your STAT program"?


I don't think so, especially if you modify it a bit:


  "My experience as a STAT major, combined with my research experiences in DS/ML, have inspired me to further study STAT."


I think you can, and should, blend the two together. As you noted, they are related, and the committees know this. Use that to your advantage.

If you are applying to a school with both and you feel like you must justify your choice of STAT over another program, then I think you've already got your justification: you are drawn to the theoretical and rigorous nature of the program. I think your major is more than enough to prevent you from seeming like an outsider.