I am trying to use pin 2 as input with a push button providing the input. The code and schematic are a bit complex as I am using 2 thru 12 as input and output. For that reason I will simplify a bit
On pin 2 I get no response. No signal nothing. When I do nothing but change the wire from pin 2 to pin 13 it works fine. This happens on 2 Arduino Uno clones that are Ruggeduino SE
Writing to an input pin effects the pull up state of and input. With your wiring you don't want an input pull up because you are using a pull down resistor. This puts you in conflict and on the limit for working. This is probably why it works on the other pins despite your error in software.
You are missing the GND connection to the pulldown resistor on your image. Therefore, your button would connect 5V to the pin when pressed, and go high impedance (open circuit) when released. Therefore, nothing will happen on most of your pins set as input.
Makes sense ... you don't need the pulldown here because there's probably already a weak pulldown for pin 13 on the PCB:
D13 LED Isolated The LED connected to D13 is isolated by a transistor circuit rather than being driven directly from the ATmega328P pin. This isolation draws much less current from the ATmega328P pin allowing it to be used for other purposes.
Therefore, the circuit for D13 (including your pushbutton) is probably something like this: