I don't know if there is a special name for cameras that don't have a wide angle view, but such a camera is not the best fit for an FPV camera.
If you do want a "small, lightweight camera without the wide angle lenses" anyway, and you don't care about the quality, look for an analog FPV camera with a long focal length. As I understand it you want to record footage as well, and depending on your needs and use cases there might be better options.
Why should you get a camera with a wide FOV, and what options are there if you want to record as well?
All the information about where you are flying comes from the FPV camera (unless you fly line of sight), so you will want to get as much information as possible from it. A wider field of view will give you that.
Most FPV cameras are not well suited for recording (high quality) footage, and most of them can't record onboard (meaning that you will have to record the lower quality footage that is sent to your video reciever). If you only want one camera, and you want to record acceptable quality footage, the DJI FPV camera is probably the best option since it can record onboard the drone. That camera (like a lot of other FVP cameras) has a FOV of 150°, so the fish eye effect isn't that noticeable. There are a few drawbacks with the DJI camera if you want to race though, since it is a digital system (See the question Difference between analog and digital FPV systems?), so an analog camera might be better if you are okay with recording low quality footage.
If you want to record higher quality footage than what the DJI camera or an analog camera provides, you will need to mount a GoPro or similar on your drone, alongside the FPV camera. This will significantly increase the weight, so you might want to avoid that, or just remove it when racing.