It is true that ozone concentrations vary considerable due to natural phenomena. For an example, look at this animation to see how stratospheric ozone varies from day to day. To look for systematic changes due to human activities, it is necessary to look at averages over longer time periods. No natural phenomenon has been brought forward that could fully explain these observations. At some point, these chemicals will be mixed back into the troposphere, where they will be rapidly removed by rain and other forms of precipitation.
Storms create ozone through the action of lightning.
Thus it may be true that some of the smell after a storm is ozone. Another possibility is that you are smelling tiny particles in the air which have been ionized in the powerful electric fields present in a lightning storm. When I was an undergrad, I noticed that there is often a strong smell around copy machines that smells like the odor after a storm. I wanted to see if the smell was due to ozone, so I made measurements with an ozone analyzer around an active copy machine. While there were some periods of elevated ozone around the machine, the concentrations were not high enough for the human nose to smell.