This is correct. I agree 100%. But ...
"...Given this information, it's best to not add borax and washing soda to the liquid soap. Add it separately to your load of laundry...."
... your premise does not support this conclusion. You are saying valid things, but you're getting things a bit garbled.
Both borax and washing soda function as alkalis when used with true soap. The pH of these two chemicals is also in the same range as soap. (see info below)
Neither borax nor washing soda will decompose true soap into fatty acids.
If these chemicals actually did cause true soap to decompose, why would you want to add them to the wash water? That doesn't make any more sense than adding them to a laundry soap product.
The point of adding borax or washing soda for washing clothes is (1) to maintain the alkalinity (high pH) of the wash water in an optimum range for soap to work well and (2) to react with and remove hard water minerals from the water to reduce soap scum. They do a good job of both.
The problem with adding washing soda to a
liquid laundry soap is the washing soda will gradually decompose upon exposure to water and the carbon dioxide in air. So adding washing soda in a liquid soap isn't a good idea if you are also going to store the product for a longish time.
Add washing soda to a
dry laundry soap mix or add it to the laundry water right when you do a load. Either of these options helps the washing soda to stay effective.
Borax is not as effective as washing soda but it is stable in a water-based product. That means it can be added to a liquid laundry soap and retain its usefulness over time. I still think it's better to add borax as a separate product to the wash water -- or better yet add washing soda to the wash water instead. That way you can adjust the amount to suit the situation. But that's more of a personal preference thing.
Baking soda (sodium bicarbonate) is another story. It is indeed alkaline, but its pH is lower than soap. When combined with soap, baking soda can function essentially as a very weak acid. That may seem odd, but it's true. This means if you add baking soda to soap, the soap may at least partially decompose into fatty acids. Furthermore it is not nearly as good at softening water nor at maintaining a sufficiently high pH so the soap can do its job well.
I don't recommend baking soda as an additive for use with true soap -- stick to washing soda or borax.
Baking soda has a pH of about 8.
Washing soda has a pH of about 11.
Borax will have a pH in the range of 9 to 10 in an alkaline mixture like soapy laundry water.
Lye-based true soap has a pH in the 9 to 11 range.
More:
Laundry soap mix | Soapy Stuff