No, that’s not hydroplaning. You are just using too much throttle on take off on wet roads. When shifting to second you are already moving.
Hydroplaning is when the tires are riding on the water and loose contact with the pavement, this happens at speed, not when standing still.
There are tires that have good dry traction, but lousy wet traction. When you do buy new tires, go to tirerack.com and/or consumer reports to research which tires have better wet traction.
You can Google hydroplaning to get a more in-depth information.