The ability of an observer to keep track of things is not relevant, and even completely computable things like cellular automata can be chaotic [1]. The common definition of chaotic dynamics [2] requires the system to be sensitive to initial conditions, mix the state space, and have dense periodic orbits.

[1] https://en.wikipedia.org/wiki/Rule_30#Chaos.

[2] https://en.wikipedia.org/wiki/Chaos_theory#Chaotic_dynamics

While there is a mathematical definition, there are uses of the term exactly described by you. For example, the Edge of Chaos (https://en.wikipedia.org/wiki/Edge_of_chaos) has a mathematical definition, but has been used in many additional areas because it so clearly seems to describe what happens as systems become so complex that they become effectively chaotic to a human's ability to order them. This edge can shift as tools improve and push back this edge.

Chaos is what happens when the torus in your dynamical phase space breaks up. Or, thought of another way; it's a deterministic process that generates random bit strings (because, no torus/periodic orbits).

No universally accepted mathematical definition of chaos exists. My guess we only at the beginning of understanding it and more theories will emerge.