Why do we choose cutoff frequency to be taken for "1/sqrt(2) magnitude reduce"?
I presume that red line I drew represents that components of a signal
  would neither increase or decrease in magnitude.


That would be a wrong presumption. The magnitude of the frequency response is (almost) never truly flat, and you would see this the more you zoom in. That's why you need to pick a convention of what exactly the "band edge" is. 

As with most conventions, it's somewhat arbitrary. Half power has the advantage that if adjacent filters have the same cutoff frequency the total power will be flat. So if you add the output of an 1 kHz high pass and a 1 kHz low pass, the energy sum of the outputs will be flat with no dips or peaks in the cross over region.

-3dB is also not the only convention either: a Linkwitz Riley filter typically uses -6dB at the nominal cutoff. -3dB also doesn't make a lot of sense for parametric EQ filters and for high and low shelve filter.

So it all depends on context and one should never automatically assume -3dB.