Oh! You sure are getting a lot of not-answers here. I can understand your frustration. Far too many people on the internet feel the need to say something, anything, about posts like yours without taking the time to actually read the question or give good information.
You are correct, your pictures show two separate ground connections. At first glance, it appears as if both of those ground connections connect to the metal body or box of your inverter. I agree with XERTYX that you should check and make sure that this is correct. Their instructions in
this comment are good. If you have any questions about how to use an ohmmeter you can ask me or them, or just take the whole thing into your nearest auto shop. They can test to see if those two connections are actually electrically connected (the meter reads zero or close to 0 ohms between the two connections).
Assuming that they are connected, the manufacturer probably gave you two separate connections either for convenience or because of some interesting regulation in the country where this device is being used. It doesn't look like it's meant for use in the United States. What with that 240 volt outlet and all. In this case, you can simply choose whichever connector is most convenient for your installation.
If they are separate (as in not connected) then you should connect the ground from the DC side to the chassis of your vehicle, as your vehicle and all the other equipment in it runs on DC, and there is less chance of any of that getting damaged by connecting a damaged/shorted AC device to that ground. You would then connect the ground wire for your AC system to the ground connection on the AC side of your inverter. However, if all you are doing is plugging one device into that outlet, or only plugging a power strip with a certain number of sockets into that one outlet, then the outlet ground is providing all of the ground connection you may need for the AC part of your system.
As XERTYX said, it looks as if this inverter is really designed for use in an off grid house rather than in a vehicle. If you can you might try to simply get an inverter that is designed for use in a vehicle. There are plenty of them out there.
Finally, just to clarify, what country are you in? Are you using AC devices meant for use in the United States, or somewhere else?