I found some YouTube video that appears to claim it's a problem with Arduino's design ... I don't know if that's true.
It is in fact, not a problem
at all. It is a consequence of an absolutely deliberate
design. An Arduino pin that is not connected or defined as an output or as "INPUT_PULLUP" is by definition, floating
- that is what it is supposed
to to; that it is a very high impedance (hundreds of megohms) is what makes it possible to measure capacitance and function as a proximity sensor.
What the op-amp is doing is accurately
reflecting the state of pin 13. Nothing more, nothing less. There is no need
to make any modification whatsoever
. In actual fact, it serves as a reminder that it is actually a good idea
to define unused ports in all sketches (used for serious purposes) as "INPUT_PULLUP" because floating inputs may cause "sneaker" currents in the input logic, mostly
harmless unless you are wishing to minimise power consumption, especially on "sleep". If however you wish the "L" LED not to be lit and you are not using it otherwise in the sketch, by all means set it as an output and LOW.
But the video seems to work. The video claims you're supposed to glue the 13 pin to the ground pin.
This is truly sad
but - this is the Internet after all. The video is bogus - insofar as it talks of this behaviour as a "problem" when it simply is not. The worst is yet to come however.
Now what is shown, is applying some liquid PVC glue to the two adjacent pins. PVC glue of course uses water as a solvent and is ever so slightly conductive. As it dries, the water evaporates and it ceases being conductive. Do you see?