Need to record videos - can I buy a PhotoCamera instead of VideoCamera?
TL;DR

If you have a camera of either type, use it. End of story. If you plan on taking photos as well as video, get a photo camera (I'd recommend a DSLR.) If you will only use the camera for video, get a video camera similar to the one you pictured.

Long Read

Video camera (camcorder):

Pros:


Design: Video cameras are designed to shoot video - they are built from the ground up to do exactly that, so all of the technology is focused on capturing video.
Ergonomics: video cameras are designed to capture video, and part of that design is the way the camera is meant to be held. Video cameras are designed to be easy to hold and control primarily with one hand. They often have a hand strap (small camcorders) or shoulder mount (large ENG-style cameras) where the user supports the camera with their right hand and the primary controls such as zoom and record are accessible with the fingers of the same hand.
Audio: Video cameras often have built-in microphones, microphone
holders or microphone ports (XLR, mini-XLR, or 1/8th inch).
Live-view video: Video cameras almost always feature both an electronic viewfinder and a rotating and folding screen, both of which can be used to view the video as you are recording it.
Accessories: Many video cameras have microphone holders, cold shoes and handles for accessories (a few cameras even have built-in LED lights). Some cameras have easy built in iris and focus controls, while some even include built-in neutral density filters.


Cons:


Stills: The biggest downside of these video cameras is they are often terrible at taking still photos, if they even do at all. There are a few reasons for this. 1) The sensor and processor are designed for taking video, so photos are often relatively low resolution (sometimes as low as 2 MP). 2) The form factor makes taking pictures awkward and often offers very little manual control.
Lenses: Most video cameras under $5000 USD have permanently attached lenses (not interchangeable), so if you want to zoom wider or narrower than the lens, you are out of luck.
Price: Video cameras are expensive! A very low end model will cost a few hundred dollars, a relatively good model will be about $2000-$3000 USD, and high end models can run as high as $100,000 (although I'd recommend a model at about the top of your budget (around $1000).


Photo camera (I will focus mainly on DSLRs)

Pros:


Lenses: The biggest advantage of a DSLR will likely be interchangeable lenses, which allow you to reach and focal length you want, assuming you don't mind paying for the lens.
Photos: DSLRs are designed to take photos. Their form factor and control placement makes this very easy. They will also shoot high resolution pictures (usually ~20-24 MP).
Audio: Often have internal microphones and 1/8th microphone jacks (but will never offer mini-XLR or XLR).


Cons:


Design: The SLR form factor has been around for many decades, since far before DSLRs offered video. Video is an afterthought on these cameras. It's added in as a bonus feature, but isn't supposed to be the primary function of these cameras. This applies to almost all photo cameras except, I would argue, the Lumix GH series.
Time restraints: Most DSLRs have a 20-30 minute video recording limit (depending on camera and quality). After this limit is up, you have to manually restart the recording.
Live-view video: Video can only be recorded while viewing as live-view on the DSLR screen. The viewfinder doesn't work. On top of that, many DSLRs (especially older ones) don't having tilting/flipping/rotating screens, making shooting at any angles difficult without an external monitor.
Price: These cameras aren't cheap, but good quality can be gotten with a $300-$400 body and a $200 lens (although I'd recommend a lens closer to $500). I'd also recommend an add-on shotgun mic (~$100-$250, depending on quality).


Please comment or edit if you think I missed anything.