Hmmm. Not sure why you think anionic surfactants are "basically lye." Examples of anionic surfactants include sodium lauryl sulfate, sodium laureth sulfate, ammonium lauryl sulfate, ammonium laureth sulfate, sodium lauroyl sarcosinate, and disodium laureth sulfoccinate. These surfactants are not remotely close to potassium hydroxide or sodium hydroxide, the two most common alkalies (lyes) used to make soap.
So let's talk about terminology. If you are talking about a cleanser made by saponifying fats and alkali, you're talking about soap. If you want to call a particular soap product by another name -- body wash, shampoo, facial cleanser, or whatever -- that's fine, but it's still essentially soap being dressed up with a fancy name. I'm going to call it lye-based soap to be abundantly clear.
So then there are cleansers that are not lye-based soap. These are synthetic detergents (syndets) such as the ones listed in my first paragraph. Unless you're talking to a person who makes handcrafted lye-based soap for washing hair, the word "shampoo" normally refers to a syndet blend of cleansers used for washing hair. Not soap. Syndets.
Only in the world of handcrafted lye-based soap making do you hear much about lye-based soap being used as "shampoo." So when you delve into this craft, it gets really confusing to know for certain what the word "shampoo" means to the other people you are talking to.
If you want to make lye-based soap for washing hair, that's fine. Many people do and claim wonderful things for their "shampoo" and if it works on their hair, more power to them.
On the other hand, you'll quickly see equally as many lye-based soap makers who have gotten bad results from using lye-based soap to wash their hair. I'm one of them. If I could successfully use lye-based soap for washing my hair, I'd do it happily, but I've learned from experience that lye-based soap badly damages my and my husband's hair.
Please do not get into the way of thinking that "if I can't pronounce the name of a chemical, then it has to be bad." I can't use soap on my hair, but I make a syndet shampoo that does work well. I intentionally make it as simple as possible. Not because I feel my products must meet some "crunchy" and "natural" standards, but just because it pleases me to see how simple I can go and still get good results.