Please don't let strangers on here change what you feel in your heart is best for you and your beloved kitty. Everyone copes with a loss of a pet in their own way. I watched the video that you listed, but there were quite a few other videos of cats dying or near death. The only thing I suggest is you don't do the actual taping, let one of the vet assistants or techs do it, so you are free to comfort your kitty as he transitions to spirit.
Before you or your mom makes the final decision to euthanize him, see what can be done about the urinating. Having had a diabetic cat -- diabetic nearly 6 years until I lost her to cancer -- they urinate a lot, it comes with the illness. Is his diabetes under control? Does he take insulin? Did you know you can diet control diabetes with just simple food changes? Please see the website below dedicated to people caring for a diabetic cat. Maybe someone on the board can give you some hints for the urinating.
But the fact that he is urinating in places other than his box could be a medical problem, such as a urinary tract infection. Cats with diabetes have lower resistance to illnesses and take longer to heal. My Kelley was prone to ear infections and I kept prescription medications to treat her when she had a flare up.
Or it can be behavioral--have you moved the location of his box or changed the type of litter? Has anything changed in the household, some moved in or moved out, has your schedule changed to where you are gone longer during the day, has there been any other major changes in the household, etc. If you are in multifamily housing (townhouse, condo, apartment), has there been a change in an adjacent residence, especially one that has pets?
The fact that the vet said he was healthy otherwise is very encouraging. Also speak with his vet for some ideas to solve the urinating problem.
If you do decide to euthanize kitty, you have my heartfelt sympathy on the loss. I lost my three oldest kitties in just 18 months (9 months apart) and I was devastated. I did not think to video them, but I took lots of pictures from every angle so that I could remember every little thing about them. Since they had all been ill and their coats were a mess, each got a bath, dried and combed before I took them to be cremated. This was something I had to do for myself and them -- they had to be beautiful.
In the last year I have read so many books about loosing pets and a common thread among the authors is that animals are not afraid of death like humans are. They know they will be returning to their spirit state and be whole and healthy again. Many animals will stay here longer than they want because they know their humans are not coping with their pending death. This poem is a good example from the Pet Loss website:
Poem For Cats
And God asked the feline spirit
Are you ready to come home?
Oh, yes, quite so, replied the precious soul
And, as a cat, you know I am most able
To decide anything for myself.
Are you coming then? asked God.
Soon, replied the whiskered angel
But I must come slowly
For my human friends are troubled
For you see, they need me, quite certainly.
But don't they understand? asked God
That you'll never leave them?
That your souls are intertwined. For all eternity?
That nothing is created or destroyed?
It just is....forever and ever and ever.
Eventually they will understand,
Replied the glorious cat
For I will whisper into their hearts
That I am always with them
I just am....forever and ever and ever.
Author Unknown