Sorry, you can NOT use that as a substitute for the front panel buttons and lights. It is the wrong type of switch.
The photo you posted shows wires held in fingers ending in five plastic connectors. Those are the normal wires from front panel items to the mobo's Front Panel header. The pair marked "Power Sw" goes to that pin pair in the header, and the wires lead to the front panel pushbutton "On / Off" or "Power" switch. It is a very simple momentary-contact switch that closes the connection between those two wires ONLY when the button is pushed. (It is really the same as a doorbell pushbutton.) When you release the button, it breaks the connection. That extra switch you bought does NOT open the connection, but keeps it closed until you turn it to the "off" position. The pair marked "Reset Sw" goes to the header's contacts so labelled, and its wires go to the front panel "Reset" pushbutton. It ALSO is a momentary-contact switch, just like the front Power switch. For both of these two switches, the polarity of how they are plugged into the mobo header does not matter.
The two connectors marked "Power LED+" and "....-" go to mobo panel header contacts with those labels. Here the polarity DOES matter. But if you plug them in backwards, all that will happen is that the front panel Power LED will never light up. If that happens, simply reverse the way these two wires are plugged in. The same goes for the pair of wires with the "HDD LED" label on them.
The switch you bought is intended for a common circuit design, but not for the way a computer does it. Its design, as shown in your first diagram, is to switch power from a source to a load and also to the internal LED indicator, and keep that power turned on until you change the switch setting. What a computer mobo does instead is quite a different design. It has a constant supply of a small amount of power from the PSU to the mobo at all times, and when it is "shut down" it actually has a few live circuits monitoring a few key spots. One of those is the status of the two open contacts of the front panel "Power" switch. If they ever are closed, the mobo VERY quickly starts up power to all the rest of the system and keeps it going by itself. It does NOT need the front switch to stay closed. In fact, it really does NOT want it closed! There is another function the mobo does when running. IF that switch contact is closed for a continuous time of 4 sec or more, it takes that to mean you want it to shut down immediately, not even doing a normal clean shut-down. So you never want to have that front panel switch closed for a sustained time! This design also makes other things possible. For example, the monitoring circuits of the mobo in the "shut down" state also watch for certain signals (IF you ask it to do this) from the mouse, the keyboard, and maybe a modem and the internet, to see if it should start up to respond to those devices. So, it is the MOBO, and not that front panel switch, that controls supply of power to all of the system for start-up and shut-down. That exta switch you bought cannot do the same job.