Is the electron's magnetic dipole moment influenced by the measurement method? The electric charge of an electron at rest is a constant value and is not influenced by the measurement instrument. The measurement instrument by itself can give more or less accurate result, but does not influence the strength of the electron's charge.
What is about the electron's magnetic moment. This moment is clearly linked with the electron's spin. Was the value of this magnetic moment corrected due to the measurement methods or due to a better theory?
The main question is: Is the electron's magnetic dipole moment influenced by the measurement method?
 A: What's tabulated by the Particle Data Group is the electron's magnetic moment anomaly,
$$
a = \frac{\mu_\mathrm e}{\mu_B} - 1 = \frac{g-2}{2},
$$
which has a magnitude of $a\approx10^{-3}$ and is currently known to about eleven decimal places.  (Note that this does not mean we know the electron's magnetic moment to fourteen decimal places: there is also part-per-billion uncertainty in the value of the Bohr magneton, and that uncertainty is increased if you try to use macroscopic units.)
Measurements of the magnetic moment anomaly seem to be based on single trapped electrons; this one is the most recent.
I haven't re-read the most recent CODATA paper to see where the $10^{-9}$ uncertainty in the Bohr magneton comes from. The Bohr magneton is
$$
\mu_B = \frac{e\hbar}{2m_\mathrm e}
$$
and its uncertainty is larger than the uncertainty in the ratio of the electron mass to the carbon-12 mass, but smaller than the best uncertainty in the value of Planck's constant.
To answer your question directly: isolating the electron's intrinsic magnetic moment from instrumental artifacts is a deeply important part of any such measurement.
