Not sure if I am quoted correctly, but in any event, injuries at home require about the same materials and procedures as those outside the home. There isn't any hard and fast distinction.

One thing about first aid. You do the best you can with what you have when you can. Hopefully you will speedily convey the victim to definitive care ASAP.

You frequently treat victims with items that aren't within your FAK. Seriously dehydrated victim need a drink of (hopefully) clean water. Cold victims benefit from dry warm clothing and maybe body heat from another individual.

In many cases you will be using all sorts of items that are never within your FAK.

For the record, sterile pads are of course highly desirable. I have never run out of them in treating a victim and I don't carry all that many, maybe up to ten or so in various configurations. Kling wrap or some sort of compression bandage is highly useful in applying pressure or stabilizing splints, but adequate substitutes can be devised. These improvisations should be part of anyone's skill set.

Be flexible in your outlook and adjust to the situation before you.