Sorry if the use of "correct" has caused confusion, I should more properly have used the term "adjust", although both are used to mean the same thing, a change or alteration.
"Distribution of error" applies to the angular measurements. In any closed traverse the sum of the measured angles equals (2n+4)right angles (for external angles) or (2n-4)right angles (internal angles). There will almost invariably be an error in this sum which is distributed in proportion to each angle.
The "adjustment by least squares" is commonly used in theodolite traverses as the probable errors in angular measurements are much less than the probable errors in distance measurement (unless you are using EDM, in which case the probable errors in distance are greatly reduced in comparison). The software should carry out the distribution of angular error to give the line bearings. It will then calculate the total closing error in latitude and departure of the whole traverse and apply partial "corrections" at each station in proportion to the linear distance between stations, without changing the bearings of the lines, to give the most probable co-ordinate values.
When traversing with dial or compass where the probable error in angular measurement would be greater than that in linear measurement the application of Bowditch's method of adjustment was commonly use. This "adjusts" the station co-ordinates in proportion to the linear distance, but also alters the bearings of the individual lines.
Regarding the "distribution of errors over the whole mesh". The primary closed traverse would be adjusted first, then the corrected co-ordinates of that traverse would be used for any subsidiary traverses begun from it and so on. And yes of course "loops" can be on the surface as you say, and commonly are when correlating to NG.