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Yes and yes.Is it meant then that the value of the field at a fixed point is a distribution (in the sense that repeated measurements won't give the same result, however the expectation value, the mean of the distribution, will be a fixed value, i.e. 0) rather than a fixed value?

Is the point, that due to the Heisenberg uncertainty principle, one cannot precisely determine the value of the field at a given point and as such its standard deviation is non-zero. So by saying that a quantum field is fluctuating is simply a handwavy statement of the fact that its value (in vacuo) at a given point in space-time is not prescisely determined (i.e. it is not well-defined) and this results in a non-zero vacuum energy?!