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lateapex911 has offered some good alternatives but if you want to install the old school radio you have and have it look like your second picture, or avoid having the "face plate look", you'll have to go with another plan.
It shouldn't be too difficult to fab a piece of sheet metal with the old style holes cut into it to fit the current DIN opening . This piece could be spot welded into the dash or secured with spring clips or some other mechanical means. Yes, you'll need to remove the vinyl and replace it with a piece that covers the sheet metal--but you'd have to do that anyway. This method would take a bit of effort, but you would avoid a "face plate look" as given by the photo examples, and you could use your current radio.