Barbecue beef, chicken and pork are best with a sauce tailored to the protein. That extends to other proteins, like lamb and goat, though few people consider them 'barbecue', since they're usually more Mediterranean or Middle East in flavor.

A pork sauce needs to be sweeter than a beef sauce and a chicken sauce (IMHO) needs to be spicier, since chicken is pretty bland. You can't cook a chicken for as long as you can beef or pork to impart smoke flavor, so the sauce often has 'liquid smoke' added to it, though I think that is often overdone. A touch is fine, but if you can obviously taste it, you've got too much 'liquid smoke'.

St. Louis barbecue is know more for its dry rub than its sauce.

Carolina sauce is very vinegary, St. Louis sauce to me seems to be close to a Carolina sauce, but is often close to a sweet-and-sour sauce with lots of sugar or honey to offset the vinegar. Memphis sauce has a lot of hot sauce or peppers in it, more than Kansas City sauce, which is usually quite thick.

Egg is a new one on me, but thickening a barbecue sauce is not. I make my variation of the "Warren's Barbecue Sauce" out of the Better Homes & Garden Barbecue book. It is similar to a Carolina sauce, but I use less sugar and more hot sauce, then I cook it a lot longer than the recipe calls for, so it has higher viscosity. I've been known to take a few cups of it, add some honey and cook it down to nearly a paste for steak sauce. I also make it in large quantity, I've been known to start with a #10 can of ketchup.

I suspect a sauce with egg in it would not store well, the sauce I make will last a year or longer if refrigerated.