Calling flights to lunar distances flights to deep space is highly inaccurate and pretentious. A reasonable name for the space at lunar distances is cislunar space. Flights to Mars and other planets are flights in interplanetary space. Flights well outside the solar system would be flights in interstellar space. (Voyager is at the edge of interstellar space.) Flights outside the milky way would be in intergalactic space. That might be getting us close to deep space.