What is the perfect definition of time constant?
Is it the time when the current through the inductor is 0.63 times of the maximum possible current through the inductor?

No. The &quot;maximum possible current&quot; is not right. It is the time taken for the current to rise 63% of the way from an initial value of current to the steady state current for a given applied voltage.

Or is it the time when the current is 0.63 times of the maximum possible current?

No. The &quot;maximum possible current&quot; is not right. It is the time taken for the current to rise 63% of the way from an initial value of current to the steady state current for a given applied voltage.

The difference between 1st's maximum i and 2nd's maximum i is... if inductor is present in parallel with souce then the current will not be same... I mean i through the L and current from source

This is jumbled gibberish so I don't know what you mean. Draw a diagram if you require a further answer. If English is not your first language then put that information in your user profile so we know that you may need assistance.
Another way of looking at it is shown below. (This is an RC charge curve but an RL curve would look the same for current rather than voltage.

Figure 1. When a tangent line is drawn at the beginning of a capacitor's charging curve, it will intercept the asymptote representing the capacitor's final charge at t = RC. Image source: Algebra Lab.