Ill-conditioning isn't a concept that depends on the precision that you use to compute the solution. "A small change in the data turns into a large change of the solution" isn't a concept that involves actual computation. Think of it as "the map data $\mapsto$ solution has a large Lipschitz constant".
The error on the input data does not come only from finite precision representation: most data you put into a computer come from physical measurements or approximation processes, and often the error implied by these measurement/processes is way larger than one part in $10^{-16}$. How many physical constants do we know up to that precision, for instance? Ill-conditioning says that you need to provide your input data with a large precision, otherwise any solution you get will be rubbish, even in infinite precision arithmetic.
Ill-conditioning per se refers only on the sensitivity to input data and is independent of the actual method you use to compute a solution. The error introduced by using finite precision along the computation (highly depending on the algorithm, including order of summation and parenthesization) is often referred to as algorithmic error or stability of the algorithm.
