Good research. So, am I correct to conclude that 34 degree line was not part of legislation, either in Congress or in the Territorial legislature?
There is a modern map (generated for the book, no doubt) in the book The Battle of Glorieta by Don E. Alberts that shows the border between New Mexico Territory and Arizona Territory going through Socorro, New Mexico. Beneath the Arizona Territory label is says in parenthesis, "As claimed by the CSA". If so, there may be some information on the boundary in the records of the Confederate Congress, but I've not searched them.
The southern part of New Mexico had definite sympathies with the South and the CSA may have simply claimed those parts of the New Mexico Territory that were in sympathy with them. Sort of like Lincoln and West Virginia.
I, John R. Baylor, lieutenant-colonel, commanding the Confederate Army in the Territory of Arizona, hereby take possession of said Territory in the name and behalf of the Confederate States of America.For all purposes herein specified, and until otherwise decreed or provided, the Territory of Arizona shall comprise all that portion of New Mexico lying south of the thirty-fourth parallel of north latitude."
This comes from the Sherod Hunter SCV site, History of the Arizona Territory