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Is this the correct interpretation?

Classical systems: You measure some state, with the measurement containing some error. Averaging the measurement error usually gets closer to the actual value.

Quantum systems: Your measurement influences/can influence the state, which can cause an error in the state itself. Multiple measurements means more possible influence.

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Yeah that's roughly it. In classical computers all errors can be simplified as being bit flip errors (0 instead of 1, 1 instead of 0). Like, power loss is a lot of bit flip errors that happened to target the bits that should have been 1. In quantum computers this simplification does not work, there is another type of error called a phase flip. Measurements cause phase flip errors. You can exchange the phase flip and bit flip bases by using a gate called the Hadamard gate. So if you surround measurements with Hadamard gates, you will see bit flip errors. The existence of gates like Hadamard is what makes it possible to see these kinds of things at all, and correspondingly its availability can be thought of as the thing that makes a quantum computer a quantum computer, instead of a classical computer.

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