Gene, There must be a lot of confusion concerning these rifles. So let me add to it. The minimal reference I have indicates the following.
Model 20A made from 1907-1922 after 1922 was designated model 20S. Also indicates 24 inch octagonal barrel and your's is obviously round.
Model 37 made from 1913-1916 similar to the model 29 which was made the same years. The model 29 was similar to the model 20 except a round barrel is indicated. No indication of barrel style on the Model 37 and no mention of a star stamp. Looks like one of those times that trying to run down accurate information on older firearms may cause more confusion than answers.
Yes, you did add to the confusion!! I think you got some of this info from either the 'Blue Book' or the 'Standard Guide to FireArms Values' both of which have exactly the same dates of manufacture for the model 37 - which is wrong. The model 37 was made from 1922 to 1930, they both say it replaced the model 29 and it did come after it, in fact after WWI. Also the Model 20 was made from 1913 to 1917 and some more again about 1922~1923 - this may have been to use up exsisting narrow recievers, you will notice in the top view pictue the 37 reciever is wider - also notice neither is drill & tapped for the Hepburn sight. The production dates were for all versions, 20 first, 20-S second and the 20-A last. The 20-S is the rarest and the 20-A the most common.
In all the pictures the model 37 is on top and it is the one with the round barrel, the 20-A is octagon. The barrel stamps are shown because they both say "Marlin Firearms Coporation" which dates them to 1922 or later, both also have the bullseye in the stock which also dates the same.
This 20-A is the only one I have ever heard of with a star stamp - but you do have to pull the stock to see so it would not be easily found. I was hoping to find others out there that were similar, with what looks like a early 37 and a late 20-A both with star stamps I was hoping to come up with at least a probable start date for its use but need some more examples.