There was a similar question a few days ago. If you are only interested in the phase angle AND your waveforms are undistorted/unbalanced (ideal case), then you would find it by measuring the time delay between V and I (nr. 3 in your questions). However, ideal cases are non-existant and approximations are almost never found in real-life, so distortions and unbalance exist, therefore the displacement factor would be measured as the time delay between the fundamentals of V and I. Think, for example, that you have a sine for V and a typical rectified RC load (as many consumers have). How would you measure the zero-crossing of I? Only by extracting the fundamental frequency component.
For this, various extraction methods are used and, depending on the level of THD which includes -- but is not only -- the displacement factor, there's only a positive-sequence extraction method, or more complicated PLL-based methods.
So, in short: displacement is the time delay between V and I. For distorted/unbalanced conditions, is the time delay between V and I fundamental components.
For example, consider the following block-level schematic, a harmonic extraction for positive-sequence components:
All it does is take the input signal, delay it by pi/2 and then multiply the results with reference signals sin/cos. The result is low-pass filtered then, the DC value will contain the magnitude information of the fundamental. The 1/z is optional, can be a Hilbert transformer or, just as well, a signal delay by a quarter period, but if you don't generate an additional quadrature, then you may need to add a gain of 2 at the output. The signals look like this:
And here is the FFT, for a better view:
As you can see, the fundamental is restored, meaning that the output can directly give you the Irms in THD formula. The denominator would be the input. This method makes measuring the displacement unnecessary and it will give you the correct power. After all, even if you can measure the displacement for fundamentals, it's redundant for power calculation in distorted/unbalanced conditions because then the THD applies.
For some reason, words don't come out as I would want now, so I hope I managed to transmit the good messages I had in mind. If not, I'll try to do so in the next answers. :)