I am not sure, i get where exactly are you going from here, but here's my attempt to make the first step a bit clearer -
Step #1A: "To proceed with step 1, I shall find the point of intersection of the internal tangent and the line segment joining the two centres.."

This would be step #2. Let's wait a little.

>but here's my attempt to make the first step a bit clearer -

This step is in a different direction. Still, you come to the same point.

The background of the method lays in the fact that the intersection points of the each pair of tangents are the two homothetic centers one direct and another one inverse of the two given circumferences. Firstly we will find these centers as follow. We call the bigger circumference W and the smaller w

1)Draw a line connecting the centers o1 and O1 of the given circumferences.

2)Draw two parallel radius R1 and r1 in each circumference. We call M1 and M2 the intersection points with W and analogously m1 and m2 for w.

3)Since we are focusing in the interior tangents only draw a line connecting M1 with m2. This line will intercept line O1o1 in a point we will call H2 which is the center of the inverse homothecy between the two circumferences.

4)Draw a circumference which has O1H2 as diameter (Obviously its center is the midpoint of O1H2). This circumference will cut circumference W in two points P1 and P2.

5)Now draw a line P1H2, this line is tangent also to circumference w. Do the same for P2 and the line P2H2 is the other interior tangent.

If you were interested for the exterior tangents in step 3) you must connect M1 with m1 and follow analogous steps.

Your idea of finding the homothety center(s) of the given circles is also valid and in fact may still end up with the construction of a tangent from a point.