Need help diagnosing oscilloscope results, newbie question about clock signals
I don't see any supply rail capacitors on your breadboard. You need to add at least 100 nF of capacitance close to each digital IC on that breadboard, located as close as possible to the IC's VCC pin.

The first circuit has a "smooth" signal with a slowly varying load and is not really affected by this (or at least you won't see how unless you compare them). When you add a Schmitt-trigger this changes. The Schmitt-trigger will draw a lot of current in a very short time in order to drive the output high and low. This will cause the voltage rails to dip and ring, and that will affect the whole circuit.