I want to take photos out from a video that I took, but would that still be photography?
If the result is a photograph, then it's photography.
A video is just a sequence of still frames, and recording video is only 1 step up from the high speed drive modes (sports modes) on most cameras.
The characteristics of the image are often quite different:
- Defects which wouldn't be visible in a moving video may become quite noticeable in stills
- Resolution is quite a bit lower in video
- Different framing techniques are often used for moving subjects.
I think you can argue that video IS photography. Take a look at the credits of films, you often see a 'Director of Photography'
In photography you can control the shutter speed (meaning the exposure time) at will (or at least allowing for available light).
In videography and cinematography the exposure time is normally set to twice the frame rate. So if you're shooting 60fps, your exposure time is fixed st 1/120th sec. This rule isn't set in stone, as that link discusses in more detail.
This is done so that motion blurring is reproduced in a way that let's the human cinema/video viewer correctly interpret it.
Now in photography you would (should) choose your shutter speed to match the effect you want.
So for some purposes a video frame won't work well as you (typically) let go of control of shutter speed.
I would say it is, in broad terms, a kind of photography. It's a way of getting a still image while using the camera for video. It's a matter of priorities.
Some would consider it cheating since a good deal of the skill of photography is in setting up the shot, which includes timing. The bigger issue will be the quality of the result - video captures don't have the quality that most people expect from still photography. A 1080p video only produces 2 megapixels! A 4k video will hit 8 megapixels, which was acceptable 10 years ago but is cell-phone quality today. In addition the compression applied to videos is more severe than the compression applied to photographs, with the theory being that you won't notice fleeting artifacts. Those artifacts will be permanently baked into the image once you take a single frame.
Those are the reasons why it isn't commonly done, but it's not the last word. If you have a video and you've taken a frame from it that makes you happy, then you have a legitimate photo.