AC current always lags voltage in a capacitor regardless of its use in series or parallel. THe impedance (f) always determines the current flowing thru it with an AC voltage across it.
But the impedance ratio as a function of frequency determines both transfer function and the phase shift relative to say a resistance where the breakpoint is 45 deg when the R=Zc(f) with lagging current then approaches 90deg depending on the position of the cap Series or Shunt and of course freq.
It is easy to visualize with Phasors and math if you understand it.
Perhaps harder is the intuitive understanding.
Transfer function of XY out vs in is shown for both HPF,LPF
and Transfer function of each part is shown XY=VI where a resistor is always a 45 degree linear line for V/I slope , but Caps always 90 deg lagging current.
Thus the position of the Cap determines if the phase of the output is partially leading or lagging when the voltage across the cap reduces near 0 above the same breakpoint. We consider Caps as short circuits intuitively when Zc<< R and thus when attenuating in a LPF it is always 90 deg. at high f and in a HPF it is always 0 deg shifted or out=in for a transfer slope depending on scales for XY looking like ~45 deg more or less IF they were equal scales.
So in summary each R and C the XY plots or Y/X= I/V on the Y/X plots and transfer function X/Y= Out/in
So I made a simulation here.
Although sweeping both filters in slow motion, they must span almost 3 decades (50Hz to 20kHz) to show the entire 90 phase shift of each filter, where it shifts most rapid at the break point f-3dB or 0.707 of input.
Note only the HPF transfer function reaches vertical or 90 deg phase shift at max f to match the phase shift of I/V in the cap when it has near 0 voltage across is.
Thus the load V(R) shows the Cap current I(C) going thru it leading by 90 deg relative to the input X or + 90deg. because the cap current is ALWAYS lagging current or Leading Voltage by 90 deg.