Thank you all for the response. As suggested, I might source a new one just to be safe and considering the hassle of changing it once installed.
@WRe, thank you for the pdf. And yes, it helps and confirms my theory of reassembly. However, it does not answer the question of placement of the single spring / plunger. I also saw in the pdf (and in my first picture) but it doesn’t show where to place it.
For now, my assumption is that it will go between the ignition housing and the switch, applying a bit of pressure to the switch mechanism for possibly contact issues.