poles and zeroes, why the magnitude plot contradicts the actual definition
What is the intuitive way to understand this ?


There is a certain amount of intuition but there's more math than intuition I would say. Below is an example I use a lot: -



The top 3 pictures show the spectral (bode) response of a resonant low pass filter (as an example) with varying degrees of peaking. The bottom left picture shows how the bode plot and pole-zero diagram combine into one 3D picture. The bottom right picture shows the view above the 3D picture i.e. the pole zero diagram.

There are couple of things that are not intuitive and this relates to where the bode plot magnitude peak occurs. If you went by where the pole projects onto the jw axis you get a frequency of \$\omega_n\sqrt{1 - \zeta^2}\$ whereas if you looked at where the peak actually occurs you would find it at \$\omega_n\sqrt{1 - 2\zeta^2}\$. This is all fairly trivial mathematically but for just a simple 2nd order system it's more mathematical than intuitive.

Same with the amplitude of the peak versus damping - the peak amplitude formula is not at all intuitive. Neither is it intuitive when you recognize that at the point in the spectrum where the phase shift is 90 degrees the magnitude of the bode plot exactly equals Q (or \$\frac{1}{2\zeta}\$).

The above is a fairly simple 2nd order example but what about a high order butterworth filter - how intuitive is it that all the poles lie equi-spaced on circle whose radius is \$\omega_n\$.


  when we draw the bode plot or magnitude plot the exact opposite
  happens


You must be getting something muddled up here.