You've just run head on into contact bounce aka switch bounce! What you think is a single button press event is 'seen' by the circuit as many contact closure events. The 555 circuit includes debouncing by design - C2 has to charge to the opposite 1/3:2/3 Vcc threshold via R2 before another switch closure can register.
Mechanical switches (and other switching contacts) bounce.
* They bounce when they close, and to some extent when they open. This can most easily be seen with the switch in series with a pullup or pulldown, across a DC voltage supply, on a DSO in one-shot mode, set to capture for min. 10ms at the highest sample rate that it can do for a 10ms-20ms capture window and the trigger threshold set to 1/2 of your circuit's Vcc. Set the trigger polarity to rising or falling depending on the pullup/pulldown configuration, and whether you want to capture contact closure bounce or contact opening bounce. Ideally, set up the DSO for 5%-10% pretrigger in the capture window, so you can see any early disturbances that fail to reach the trigger threshold. A logic analyser will be less informative than a DSO as it conceals the analog behavior of the switch output.
if you set up a CMOS counter IC with LEDs (+ resistors) on all outputs. and clock it from a mechanical switch as in your 4013 circuit (or if you've got some more 4013 chips, wire them (+ the spare flipflop in the existing one) to toggle (/Q to D) and clock each from the previous stage) you'll see it increment several to many times for each switch press.
A classic method of implementing a fully debounced switch for clocked logic experiments is to use a break before make SPDT switch, common to the rail, with NC and NO contacts feeding the S and R inputs of a SR flipflop, with pullups/pulldowns to keep them in their inactive state while the moving contact transitions. Which rail (Vcc or ground, the common connects to depends on whether the flipflop has active high or active low inputs. For classic TTL, its almost invariably ground, as the current required to reliably pull a TTL input low is not insignificant, so pulldowns are generally impractical.
For more details see:
https://www.electronicdesign.com/technologies/analog/article/21155418/logiswitch-11-myths-about-switch-bouncedebounceand for mitigation methods see:
http://www.ganssle.com/debouncing.htmwhich is *essential* reading for any tech or engineer who dabbles in logic or embedded programming.
* *SOME* switching contacts do not bounce by design, however mercury wetted contacts (which are immune to bounce if properly designed) are now vanishingly rare, and elastomer contacts substitute ramping contact resistance for switch bounce, which when applied to a logic input without hysterisis, has much the same effect due to noise toggling the input as the switch resistance change sweeps it through its threshold region.