distances - What is the definition of "deep space"?

It's not in any sort of academic context, but in relation to space exploration NASA defines deep space as such:

Deep space is the vast region of space that extends beyond our Moon, to Mars and across our solar system.

Again, not exactly an academic context, but Govert Schiling (the author of Deep Space: Beyond the Solar System to the End of the Universe and the Beginning of Time) in relation to astronomy defines deep space differently:

In a sense, everything beyond Earth’s atmosphere is deep space. But more typically it’s a phrase that has been used by observational and amateur astronomers alike for extended objects beyond the Solar System. I decided to provide a brief introductory section on the Sun and the planets, but the emphasis is really on the objects in our Galaxy and the wider Universe.

So this more or less echoes what some others have contributed, but provides additional context.