Phase Shift due to Capacitor
"Voltage lags current by 90&deg;" is correct &mdash; but you never actually measure the current through the capacitor. Try adding that measurement to both of your experiments, and you find that this is always true about the voltage and current associated with the capacitor.

In your second circuit, you've essentially turned the voltage source into a current source (by making the resistor so much larger than the capacitor's impedance), so now you can see the phase lag.