I thought DeepResearch has the AI driving the process because it's been trained to do so vs DeepSearch is something like langchain + prompt engineering?
DeepResearch (note the absence of the space character) is the name that Han Xiao proposes for the general pattern of generating a research-style report after running multiple searches.
You might implement that pattern using prompt engineering or using some custom trained model or through other means. If the eventual output looks like a report and it ran multiple searches along the way it fits Han's "DeepResearch" definition.
I guess they will take any opportunity to follow the leader here if they worry that they're at risk of similar branding issues here too.