It is JATO because it is a "Jet of hot expanding gases", whatever if the device used is a "Rocket" engine, a "Turbo Jet" engine or a "Stato Jet" engine; from the combustion of solid fuel or liquid fuel.
English Wikipedia defines "Rocket" as: "A rocket engine, or simply "rocket", is a jet engine that uses only stored propellant mass for forming its high speed propulsive jet".
In other words, a "Jet" engine is NOT ALWAYS a "Turbo Jet". It can ALSO be a "Rocket", or a "Stato Jet".
Be aware that I'll remove any offending comment and block it's author immediately!
A most impressive Jet Assisted Take Off (JATO) take-off at Miramar NAS Air Show, on August 16, 1997
Even more impressive when on big planes like this C-130. It's "Fat Albert" of the Blue Angels. This was during one of the very last Miramar Air Shows with the Navy. It was in the early part of the Twilight Miramar Air Show.