'actual' or 'true' randomness is a rabbit hole.
https://en.wikipedia.org/wiki/Randomness
It really depends on the exact definition of what you are quantifying 'random' to be.
There is no proof (in the mathematical sense) of real randomness.
I think that makes sense and I can tell that you understand the distinction. Formally I believe they're different concepts and think it may cause confusion in some cases to use them interchangeably.
The chaotic nature of a system is one thing.
Our lack of knowledge of the governing laws, initial conditions, feasibility of simulation forcing us to use the mathematical tools of probability (i.e. randomness) to describe our uncertainty about said system is another thing.
The reason why it matters is that a statement like "a double inverted pendulum behaves randomly" is just wrong as it would imply that you couldn't even do a simulation of one in theory without throwing some dice.
However, it is totally uncontroversial that if someone gave you a measured initial position and velocity of one with 'really good' precision and asked you to predict its state 5 seconds forward you would likely have a big smeared-out probability density function to deal with.
There are a number of sophisticated tests for randomness. You can't prove absolute randomness in any Platonic sense, but you can certainly assess a source for different properties that are useful in applications that require randomness.
In this example the path is neither chaotic (in the formal sense) nor random, because a Fourier transform would identify the harmonic components.