Voltage drop in RC filter
Both correctly describe how an RC filter will react at DC. The reason you see a contradiction is because you're actually describing two different situations: loaded, and unloaded. If there's a load, there's current flow from the source, through the resistor, to the load. The voltage drop across the resistor will be \$V=IR\$. If there's not a load, there's no current flow, so there's no voltage drop across the resistor.

Now, when you say "at low frequencies", as long as there's any frequency content, the capacitor will have some effect on the output. But for DC, you seem to understand the analysis correctly.