Linear aproximation of an inductor in a switch circuit
You're mixing models here -- you need to get down to the actual behavior of an inductor.

When the inductor is connected on DC power supply it will build magnetic field which induces voltage that opposes to the source voltage, which than produces current that opposes the original current. So the original current through the inductor...

This is a handwavy way of saying that an inductor self-induces a voltage.  Yes, all that happens.  As a consequence, the current on a perfect inductor follows the equation $$v(t) = L \frac{d}{dt} i(t)$$
That means that for a perfect inductor, and for a constant voltage, the current will rise in a perfect ramp.  You can solve the above differential equation and see that.

However, I have trouble to fit this concept into the physical picture of the inductor knowing that the transient changes on a inductor are described with an \$i(t) = I(1-e^{-t/Tau})\$ and \$i(t) = Ie^{-t/Tau}\$.

That is true for an inductor in series with a resistance with a constant voltage applied.  In the case of switching supply analysis, the resistance of a typical coil is small enough compared to the product of the switching time and the inductance that it doesn't matter much to the analysis.
In a well-constructed supply, long before you need to consider the effects of resistivity on the coil's current &quot;shape&quot;, you need to consider non-ideal (and nonlinear) effects in the inductor core.  At that point you're done drawing straight lines on paper, and you need to take your (hopefully very) educated guess about the power supply design and simulate it.