What I would consider design research is more foundational. Typically done at the upfront of a design process (hardware or software) and will involve some or all of the following:
- ethnographic sessions
- retail intercepts
- online user review synthesis
- professional review synthesis
- aesthetic trend scraping
- tech trend scraping
- category/industry trend scraping
A good design research output will synthesize all of these inputs into a concise set of recommendations or insights in an executive overview. That deck might include key opportunity areas and use cases for a potential product. This will then feed the hardware/software design and engineering teams when exploring archetypes, feature set, product specs etc… that then will feed into the product design process (hardware or software)
I consider UX research to come after that when refining use cases, then later in the design process to test interaction models.
To put it not suscinclty, design research is more broad (it can also be more expensive and more easily skipped, a fully immersed team can do most of it on the go, gurilla style). UX research tends to be more tactical, testing based, and empirical. Those are broad statements of course, the line between them blurs and people missuse terms all the time.