I understand your confusion regarding the marking system and specifically the marking of the word "author." Let me explain.
The marking system devised by Romalda Spalding helps children learn to "spell" words (not just pronounce words). Pronunciation varies greatly from region to region within the United States and certainly within English speaking countries around the world. Relying on pronunciation alone soon gets one into trouble with spelling. Remember that spoken language is ever changing while the way we spell words remains relatively constant.
The Spalding Method is based on the text, The Writing Road to Reading. This is not just a play on words; it truly is the way children learn to write, then read. Children first identify the sounds in words, then associate phonograms with those sounds, then write the words, and finally read the words they wrote.
To ensure students are hearing the sounds that are needed for correct spelling, some vowel sounds or vowel combinations are strongly emphasized. For example, the word "author." If you say this word in the normal rhythm of speaking, it does sound like "auther;" however, you don't want that spelling so you clearly and plainly pronounce the word (for teaching spelling) as it is actually spelled. You say "au thor." After the child has written the word (we hope correctly), he/she reads it in the normal way of speaking. The marking emphasizes the sounds needed for correct spelling.
You mentioned encountering other such examples where marking does not seem to match pronunciation. This is very true. One example would be the word "come." If you say this word in the normal way we speak, and the child wrote what he heard, he/she would probably write "cum." You do not want this, so you must clearly emphasize the sounds needed for spelling so the child will write the word correctly. When the child reads the word, he/she says come in the normal way.
Over time, speech has deviated from spelling, making it harder and harder to figure out how to spell words from the way they sound. Mrs. Spalding developed her marking system as a means of counteracting this difficulty.
Hope this helps.