January 7th 2014 Edit
I currently have a PCIE x4 Intel I350 NIC in a PCIE x16 slot meant for graphics on a Intel DH77DF motherboard working without a problem.
The original post below stating that "as long as it fits it should work" still stands. It's still possible that some motherboard manufacturers have a PCIE x16 slot that won't accept anything other than a graphics card. I think this only happened in the early days of PCI Express, and modern motherboards (say Core 2 and newer, and definitely Core i-series and newer) are pretty flexible in what you can use the PCIE x16 slot for.
Also, make sure you have on-board or on-chip graphics, or a video card somewhere. Most motherboards won't boot without video. Although, sometimes you might be able to configure the BIOS to ignore the lack of video and continue booting.
I figured I'd return to clarify my answer after encountering this very situation myself :)
Original Post
Initially I'd say there's no real difference between the PCI-Express slots and as long as it fits there's no problem.
However, the problem is if the PCI-Express slot is specifically labeled for video card. The tomshardware link shows the user placing the TV Tuner card in his second slot, not the first. The first slot (in your case, your only slot) might be graphics only.
Some point to yes. Some point to no. It's hard to tell and seems like it's up to how the motherboard manufacturers decided to integrate the chipsets.
What you want is a riser card like this. The problem is that almost all standard cases won't accept the the add in cards, especially if they have external interfaces, unless they were designed with them in mind. Such cases are typically 2U rack mountable. Also, with this particular card, there may be other clearance issues since the cards point back toward the CPU.
Best Answer
The orange slot is PCIE 16x, the small one is PCIE 1x and the larger white one is PCI. You can get a cheap video card that will fit the PCIE 16x. Most of the cheaper PCIE 16x cards will run you about $50 and have DVI, VGA, and HDMI ports. You would use the VGA port to connect to your projector.
By the way, a lot of cards you have posted there are AGP (old school connection) and PCI which would fit your larger white slot. However, since PCI is a bit older, you'll often find that they have a similar price to the newer and cheaper PCIE 16x cards. Not only are they slower and older, some newer computers might not have a PCI slot and therefore you wouldn't be able to salvage your new card into a new computer.
You can tell that the orange slot is not AGP for the simple fact that you have the tiny PCIE 1x slot there. They never made a PCIE 1x slot on the same motherboard that they did the AGP slots. There is a 5-10 year difference in the technologies.