In my view, design research is the scientific activities which aim to generate knowledge to solve a problem in a product, service or system design project. One design research methodology I've learnt called research through design, which you can find more information with the link below. This type of research is mainly in design descipline.
Research design is the scientific methodology which you can follow to define the key elements and process of your research project. It's applicable to most desciplines. Herewith a classic textbook for research design.
John W. Creswell. Research Design: Qualitative, Quantitative, and Mixed Methods Approaches
Aryan Shahabian In my Design Research project (VR facilitated ergonomic design), I used Research Design method to construct my studies. In my view, research design is a general approach to plan studies in a systematic manner, which belongs to the scietific philosopy, so it can be applied to different studies, like mathemetics, informatics, engineering, social science. Anyway, I'm not an expert about both concepts, you'd better ask experts, e.g.
Design Research - Prof. P.J. Stappers proposed Research through design method,
Research Design - Prof. S.D. Zwart teaches Research Design course in my university, https://www.tudelft.nl/tbm/over-de-faculteit/afdelingen/values-technology-and-innovation/people/assistant-professors/dr-sd-sjoerd-zwart/