I may be able to shed some light on your experience.
Eye Docs always check for astigmatism, half the population of the world have a small amount of it. Albinos tend to have a lot, maybe between 2 and 6 Diopters.
When you squeeze your cornea you do change your astigmatism - but in an uncontrolled way. You found a certain measure of success when you pressed in a certain way- but there is no guarantee that the distortion you have produced is symmetrical. What you have discovered for yourself is irregular astigmatism in which the curvature of one side of your eye is not the same as the other side Pressing one side of your eye more than the other side tends to even things out.
Regular glasses are able to correct regular symmetrical astigmatism but do not correct irregular astigmatism. There have been attempts to correct irregular astigmatism in glasses but success has been limited.
It is only recently that the eye Doc has had the instrumentation (abberometer) to measure irregular astigmatism. These instruments are really the province of the lasik surgeon.
You may consider it worthwhile to have a lasik surgeon check the curvature of your eye. You might then consider lasik surgery, not to fix your overall refractive error but merely to trim your astigmatism back to symmetry. At that point regular glasses should offer a better improvement to your vision.