Where did the base get the current from?
All transistors leak - maybe much less than a microampere, but more than zero.
Your transistors are not different. You may short the CB leak with the switch, but when you release the switch the rightmost transistor gets some base current as CB leakage ==&gt; the leftmost transistor surely gets some base current through rightmost transistor. That makes the leftmost transistor conducting and outputting base current to the rightmost transistor. It's like a flipflop. You have the reset switch. There's no set switch, but CB leak autotriggers it.
You probably find a resistor which in parallel of the switch eats enough the CB leak and the flipflop doesn't get autotriggered as soon as you release the switch. I would try at first 3300 Ohm.
The set -switch would connect via a resistor either the base of the leftmost transistor to GND or the base of the rightmost transistor to +5V.
IMPORTANT: There should be a resistor somewhere in the path from +5V through the transistors to GND - preferably between the base of the PNP tr. and the collector of the NPN transistor. I would insert at least 2 kOhms. Otherwise the transistors can burn due no current limit.
BTW. No leak probably would be even needed. The sparse wiring collects easily enough noise and hum to trigger the circuit. Small enough resistor in parallel  with the switch eats also noises that the wiring collects.
ADD after the edit of the question: The mosfet version behaves oppositely when compared to the original circuit. The collector current of T2 reduces the drive of the N-Mosfet T1. The results with the original circuit are useless with the mosfet version.