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The U.S. government defines deep space as anything beyond low-earth orbit. [1] Others vary. [2] The U.S. "Deep Space Network" was created because communications with low-earth orbit and beyond-LEO had very different requirements [3].

From the NASA Deep Space Communications Educator guide [1]:

> Deep space begins at approximately 42,000 km (26,098 mi) from Earth

[1] https://www.nasa.gov/sites/default/files/atoms/files/ps-0343...

Deep space should be the furthest point reached by probes in likeness to deep ocean depth exploration. Voyager I, II and New Horizon probes should redefine the ”distance.“

I believe they call that interstellar space to distinguish from "deep" space.