Do I have a serious electrical wiring issue?

You need an electrician to sort this out, as you have probably tripped the breaker or blown a fuse.

You don't know enough to solve this if someone else wired the rest of the circuit wrong.

The neutral wire is not working and someone used the ground to "make" the receptacle work. It's not correct and as you see not particularly safe. That's why when you put the wires where they are supposed to be the outlet stopped working.

Turn off the power and go box to box inspecting the wires, splices and connections at all other outlets on this circuit. Hopefully you will find and fix the reason the neutral doesn't work. If your outlets have the wires pushed into the back, pull them out and wrap them clockwise around the screws.

If after that the neutral still doesn't work then the wire is damaged somewhere between boxes and fixing that can be anything from moderately difficult to impossible without removing baseboards, crawling under the house or through the attic.

The diagram you show is for a split receptacle where one outlet is hot at all times while the other is controlled by a switch. This is why the tab on the right has been removed. The tab one the left is retained in order for both outlets to share the neutral wire. Since you say that the red wire is not present in your case this does not apply. If you only have two conductors and a ground coming in to the electrical box, this is the end of a circuit. The bare ground wire should be attached to the ground terminal. The white neutral wire should be attached to the terminal with the silver terminal and the hot black wire should be attached to the darker or brass colored terminal. Once this is done the receptacle should function properly. If it does not, replace it.

It sounds like you need to replace the outlet and before you are done you will need some special tools. A receptacle tester about $5 US and a screwdriver and some wire strippers or a knife to remove the insulation. Do not use a back wire type. The device wiring indicates to me that you have an open neutral in the circuit and it may be in another outlet that needs to be inspected. The receptacle tester will help. you locate the problem area. The wiring is seldom damaged in the wall rather usually in a device box.

It sounds like there is power on the ground wire you can try to use a grounding adapter to break the ground path to see if this helps if that fixes the problem then call a electrician or fix it if you know how. The common and ground should be a separate connection.