Why phase is the position of the waveform relative to time 0
Definition:


  waveform
  /ˈweɪvfɔːm/
  noun PHYSICS
  a curve showing the shape of a wave at a given time.
   Oxford Dictionaries.


I think that Oxford have it slightly wrong and that it should be "a curve showing the shape of its graph as a function of time". Wikipedia is better.

The waveform represents the variation in a value with respect to time. This could be a voltage (electrical), a height (wave on the sea), pressure (sound), etc. and simply shows the instantaneous value at any point on the time axis.


  According to the picture, at time 0, the waveform is just a point. 


Not quite. We can deduce two things from the graph: 


At t = 0 the amplitude is zero.
At t = 0 the amplitude is rising at a rate given by the slope of the curve at that point.



  Later it starts to form a waveform shape, so how come a waveform is relative to a point at time 0?


The diagram is comparing the relative phase of three different waves of the same frequency with a periodic time of T.


The first has been taken as reference and t = 0 has, for convenience, been chosen as the point where the wave crosses the time axis. (It's easy to visually see and compare the intersection of the curve with the axis.)
The second is is at maximum at t = 0. By sketching in the dotted section we can see that it crossed the time axis earlier than the first trace and is leading the first by 90&deg; or T/4.
The third trace is leading by 180&deg; or T/4.



  ... so how come a waveform is relative to a point at time 0?


Typically t = 0 is just chosen for convenience to make the illustration clear.