Info Request: Early Winchester semi-auto

There were three early Winchester semi-auto rifles, all similar.
There was the Model 1905, available in .32 Winchester and .35 Winchester.
It was made from 1905 to 1920, with 29,113 made.

The Model 1907 was available in .351 Winchester.
There were 58,490 made between 1907 and 1957.

The Model 1910 was available in .401 Winchester.
It was made from 1910 to 1936, with 20,786 made.

The Model 1907 was very popular with lawmen in the 1920's and 30's due to the powerful .301 cartridge and the semi-auto operation.
Many were sent to a custom gunsmith in Joplin Missouri to have extended box magazines fitted.
Famed Texas Ranger Frank Hamer had one built when he took up the pursuit of Bonnie and Clyde.

Values for the various models in 90% condition are "ABOUT":
Model 1905- $550.00
Model 1907- $450.00
Model 1010- $550.00

The Winchesters don't have a shrouded barrel though, the Remington model 8 did. I think it's what you are describing. Square back (Browning collaboration on design), shrouded barrel, and sidelever safety...

http://www.remington.com/library/history/firearm_models/centerfire/model_8.asp

Sounds like a Remington

Perhaps the Model 8:



These were a Browning design, long recoil operated, in the Remington rimless calibres (.25, .30, .32, .35), introduced in 1906 and replaced by the similar but updated Model 81 in 1936 (discontinued 1950):