What you have said is partially true, but perhaps some clarification here might help you work out where to draw the line.
User Experience Design is a particular approach to Design that is an amalgamation of methodologies and techniques from a number of different disciplines.
Design Thinking is a particular approach to Thinking that has been developed to enhance design related activities, which also combines a number of different concepts and methodologies.
Therefore, it is not uncommon to see people who are UX designer employ design thinking techniques, or people who are design thinking practitioners involved in UX design activities. The difference probably comes down to semantics or how you would like to draw the Venn diagrams, but essentially they are not really constrained to a particular field or discipline because they are the implementation of abstract concepts to help with a particular approach to do something.
In fact, you will also find many similarities and overlaps if you look at many of the tools and methodologies used in Agile methodologies and practices, because they are also aimed at solving problems in a similar way or in related domains.
So when you commented that
I've been hearing a lot of talk about "Design Thinking" as a "thing"
but every time I look up information on it, it seems to be exactly
what we do in UX design but just in other areas.
It is quite true, and you'll find people who are very knowledgeable practitioners of Agile methodologies also doing very similar things (aside from all the other activities related to software development rather than design).
However, the major difference between the two is that UX Design also incorporates many aspects of the design process (in terms of an actual product or service) whereas design thinking is aimed at more of the conceptualization and research activities in the earlier phases of UX Design.
I hope this helps to make some distinction between what can be conceived as two very similar concepts and approaches used in modern software design and development.