From what I've seen, the badge differences are often for special runs - like Centenial, and the Silver that is only on the 1932/1933 updated 15 models.
On the back of the early small green manuals (the small ones started coming out around 1913)
"This Trade Mark Embossed in Brass is on the Arm of Every Singer Sewing Machine".
Before that manuals were larger and tan, and then a few large and green, then the small green up until about 1953 when they started coming in different colors and sizes
In 1936, the manuals where changes and the "Embossed in Brass" is no longer there.
in 1947 the back was changed to a white square of information - and refers to the "Red S" as identifying authorized Singer Sewing Centers
I think it's funny when sellers tout "red S" as if it was something special. All it means is the machine is a later made one.
Some other interesting notes
1929 - the logo on the manuals changed from treadle to electric
1947 - logo changed to black shoes and machine( was outline before that)
1948 * was added above the logo indicating registered trademark - name on front changed from Company to Co.
How do I know this? because I've made a study of small green Singer USA manuals that are for sale on eBay
So far I've collected info on 22 different ones for the 66, 7 for the 99, 21 for the 15, 3 for the 101, 9 for the 201, 10 (and probably more) for the 221 (note - the 222 came out after the small green manuals were discontinued). 7 for the 127 (27 was discontinued at the time the manuals were changed so I've seen no small green ones for the 27)
10 for the 28/128, 3 for the Electric motor conversion.
There are at least 18 more manuals - as I have Revisions of a number, but have not located one to see what the original publication date is. So that makes at least 115 different ones. I already own 23, so only 92 more to go. Not sure if I want to actually collect them - even at $10 each that would be a big chunk of money. But it would be a fun thing to do!