How to calculate the kinetic energy spectra from ocean current timeseries?
Do you get much different results if you do both ways?

I think if you want the kinetic energy spectrum, use your second method. Some separately compute velocity and KE spectra, e.g., figure 2 of http://journals.ametsoc.org/doi/pdf/10.1175/2009JPO4330.1 (DP Wang et al, JPO, Apr 2010). Comparing velocity and KE spectra, the KE spectra looks similar to the more powerful velocity spectrum, especially at low frequencies.

Others use your first method (e.g., http://journals.ametsoc.org/doi/pdf/10.1175/JPO-D-15-0087.1, Rocha et al, JPO, Feb 2016)