In Hatcher's book, the torus as a CW complex is constructed so:
But as far as I see, I can follow the gluing instruction also in the following way.
I draw the vertex $p$ and the edges $a$ and $b$ so:
Then I draw this picture on a big sphere, and the big region of the sphere that is out of these small disks will be the 2-cell. As far as I see it is glued as is required for beeing a torus (these small disks will be the two holes of the torus). Is this construction really good? Is it really a torus, or I missed something?
The construction that you did does not lead to a torus. As mentioned by John in the comments what you get is homotopic to $S^2$ minus two points which is homeomorphic to a cylinder. You can then quotient out the cylinder to get a torus.