The problem with the word "rocket" is that it is a colloquialism, subject to use by non-experts and not an exactly-defined term.
The term "Rocket" is used for any vehicle that's propelled by one or more rocket engines.
The Cambridge Dictionary is wrong. Rockets are typically cylindrical, but not universally. Famous counterexamples: N-1, V-2.
1a : a firework consisting of a case partly filled with a combustible composition fastened to a guiding stick and propelled through the air by the rearward discharge of the gases liberated by combustion
b : a similar device used as an incendiary weapon or as a propelling unit (as for a lifesaving line)
2 : a jet engine that operates on the same principle as the firework rocket, consists essentially of a combustion chamber and an exhaust nozzle, carries either liquid or solid propellants which provide the fuel and oxygen needed for combustion and thus make the engine independent of the oxygen of the air, and is used especially for the propulsion of a missile (such as a bomb or shell) or a vehicle (such as an airplane)
3 : a rocket-propelled bomb, missile, projectile, or vehicle
You can see this is very broad.
NASA uses this definition too, depending on the audience for a particular article. Note that the linked article is aimed at children.
The word "rocket" can mean different things. Most people think of a tall, thin, round vehicle. They think of a rocket that launches into space. "Rocket" can mean a type of engine. The word also can mean a vehicle that uses that engine.
and yes, according to the linked article, NASA considers the Space Shuttle to be a rocket.
Wikipedia also agrees: a rocket is any vehicle powered by a rocket engine, and it includes things like rocket cars.
In the industry, the terms "launcher/launch vehicle" and "spacecraft" are used instead. These more accurate terms avoid the ambiguity of the word 'rocket'.
The important part of the definition of a 'rocket engine' is that it does not need to draw its oxidiser from the surrounding air, which means it can function in a vacuum. This is usually done by carrying the oxidiser in a tank, in addition to the propellant. There are edge cases, but that's a subject for another question.
M-W definition of a vehicle:
1 : a means of carrying or transporting something planes, trains, and other vehicles : such as
a : motor vehicle
b : a piece of mechanized equipment