Okay, you don;t need to know every part name. The main ones you need to know are these:
[working from the manual] The 'cutting cylinder' is often called the reel (I prefer 'reel') The 'cutter bar' is often called the 'sole plate' or 'bottom block' Attached to this block by screws is a 'fixed blade' or 'bedknife'or 'bottom blade'
Now, I've looked at the video. The most obvious thing that really stands out from you photograph(s) is that the sole plate and its bedknife is touching on the ground. This is a big no-no. In your photo (reproduced below) the height is set to the absolute minimum.
Go back to the manual and set the height to at least half way by loosening each knob on the roller brackets. Your machine is skidding along the grass. The sole plate should not touch the grass at all. The back roller should.
The next step is then to post a close up of the reel and sole plate taken from underneath from the front of the machine. We need to see the condition and thickness of the bottom blade or 'bedknife' and reel or 'cylinder'
I have included a YouTube video of a similar but modern machine. Please note the concept of adjusting the reel to bedknife (discussed in your manual).