[Math] Fano plane drawings: embedding PG(2,2) into the real plane

Question

By a drawing of the Fano plane I mean a system of seven simple curves and
seven points in the real plane such that

• every point lies on exactly three curves, and every curve contains
  exactly three points;
• there is a unique curve through every pair of points, and every two
  curves intersect in exactly one point;
• the curves do not intersect except in the seven points under
  consideration.

The familiar picture

(source)

does not count as a drawing, since the last requirement is not satisfied:
there are two "illegal" intersections. In fact, this is easy to fix:

(source)

However, this drawing is degenerate in the sense that two of the curves
just "touch" each other, without crossing, at some point. And here,
eventually, my question goes:

Is every drawing of the Fano plane degenerate?

(Although I can give a topological definition of degeneracy, it is a little technical and, may be, not the smartest possible one, so I prefer to suppress it here.)

Answer 1

Does this one work?

 (source)