Think of sparging as rinsing and diluting. You sparge to dilute the concentration of sugar in the wort that remains absorbed by the grain. The higher the sugar concentration in the retained wort, the less sugar gets into your BK, so leads to lower lauter efficiency. Mash efficiency equals conversion efficiency times lauter efficiency.
If you just add the "sparge" water to the mash before running off any wort, then the retained wort has the same sugar concentration as the collected wort. If you drain all the wort from the MLT, and then add the sparge water, stir (or recirc) and run off again, then the retained wort has a much lower sugar concentration.
Let's do a numerical example: We'll use 8 lbs of grain, 8 gal of total brewing water, assume 0.125 gal/lb grain absorption, assume 100% conversion efficiency, and collect 7 gal of pre-boil wort. For the full volume mash (no sparge) we have the following:
Wort absorbed by grain = 8 lb * 0.125 gal/lb = 1.0 gal
Wort collected = 8 gal - 1 gal = 7 gal
Pre-boil & retained wort SG = 1.0335
Lauter efficiency = 82.4%
Now if we strike with 4.5 gal and sparge with 3.5 gal we have:
Wort absorbed by grain = 8 lb * 0.125 gal/lb = 1.0 gal
First runnings collected = 4.5 gal - 1 gal = 3.5 gal
First runnings SG = 1.0568
Second runnings collected = 3.5 gal (grain already saturated, no additional absorption)
Second runnings & retained wort SG = 1.0171
Pre-boil SG (combined runnings) = 1.0370
Lauter efficiency = 91.1%
Both no-sparge and sparge will make beer, there is no "correct" way to brew. What's more important than your actual efficiency is the consistency of your efficiency. You need to be able to predict your efficiency in order to predict the OG from your recipe. How you want to proceed is up to you.
Brew on