So in other words, the Schottky have a lower resistance than the 1N4001's?
No.
A diode lets current flow one way and not the other. But in the forward (conduction) direction the voltage has to be above a certain level before it will start to conduct. Think of it like a wall and water on one side has to get above the wall before it flows over top. (not a great analogy but good enough)
Silicon diodes typically start to conduct at 0.6 volts. The Schottky diode is a different technology and the conduction can start at about 0.2 volts. With more current the forward voltage goes up.
Specifications are sparse for the ESP8266. I couldn't find details for the reset input but I did for the IO ports. As an input a voltage should be at or below 0.25*Vcc. If Vcc was 3.1 volts then we would need a low to be 0.775 volts or lower. A silicon diode with a voltage drop close to 0.7 volts will give us a low if the cathode end of the diode was at 0 volts. But if there was aditional voltage drops in the path we may not have the processor see the low as a low. And that is what you were seeing with the 1K resistor. That is why we want to use a diode with a lower voltage drop.