How does one know if a well pump is "Short Cycling" or pressure tank is water logged?
At 2.5 GPM, your 2-3 minutes of shower time between cycles is using between 5 and 7.5 gallons.  If your tank starts at 60psi and 7.5 gallons is enough to bring it down to 40psi, that means you're starting out with 15 gallons of air in your 40 gallon tank, and ending the 3-minute cycle with 22.5 gallons of air.  I calculated this using Boyle's law, or P1 * V1 = P2 * V2.  And of course these numbers would be different if your shower head is higher-flow and your 3 minutes of showering used more than 7.5 gallons.

I believe to get the most output from your tank, you want it to be nearly empty of water when the pump is set to come on.  That means draining and charging it to just below the pump setting, as you did.  If you wanted to get a little more cycle time you could probably even go as high as 38 psi.  Note that with a bladderless tank, you're going to need to recharge periodically as the air in your tank is gradually absorbed by the water.

For what it's worth I think 2-3 minutes is well within your pump's design cycle time.  15-20 seconds was way too short and by noticing it and fixing it you've probably saved yourself an expensive pump replacement.