How else would you complete the circuit? Doesn't every switch have a positive and negative running to it? In this case, pressing the bell switch does sound the bell for as long as I press the switch.
It would help us to help you if you were to up-load two snapshots showing the four terminals on the two switches: one picture for the push-button switch used to ring the doorbell, and the other picture for (presumably) the toggle switch used to turn the porch light on and off. Perhaps with the aid of those two pictures someone here could identify just what your switches are and perhaps their contact configuration. Your switch "block diagrams" show four terminals on each switch with nary a clue as to how these terminals are connected internally to each switch. Zero information there.
Both push-button doorbell switches and toggle switches for lighting are usually single-pole, single-throw, two-terminal switches. You should Google that to see exactly what it means.
As for "how else would you complete the circuit?" it appears to me that you have through trial and error found a couple of ways to do that, one of which works and one which results in arcs, sparks, perhaps a little smoke, and a tripped circuit breaker. However, I see from reading other posts made while I was composing this one, that you "solved" your problem by adding a fourth wire, as shown in post #9, to your original concept shown in post #5. This you did despite saying "I only have 3 wires to work with and I can't tell the difference between a neutral or ground." Don't you think it was a bit disingenious to change the problem from "how do I do this with only three wires?" to "look how smart I am to solve this problem by adding just one more wire"?
"Doesn't every switch have a positive and negative running to it?" No. Switches are used to interrupt or complete circuits. Whatever you think you mean by "positive" and "negative" has absolutely nothing to do with switches.
And finally...
I said that I don't know what I'm doing. You don't post questions here if you know what you're doing.
Well, at least you admit you didn't know what you were doing. So many people post questions here who firmly believe they
do know what they are doing... mostly. They believe they just need a little help. In case you didn't notice, this is a "hands on" do-it-yourself hobbyist forum mostly. Most folks come here for the same reason you did: to obtain practical advice on how to solve a problem. We are happy to help with that, but we do expect a modicum of knowledge, and perhaps some experience from the poster, to allow a meaningful dialog to occur. Lacking that, I suggest you try Google first before engaging ElectronicsPoint forums.