Here's some detailed pictures of the Model 165. It has the same slot underneath the stock that yours has, but there are differences. So maybe they did make different variations with a sporter barrel and slightly different features over the years.
There’s multiple H&R magazines at Numrich with the same part number except the last digit is one of a series of letters, A, B, C, D, E, F, G, H. This may be an indication that there were several model designations of models each with slightly separate features. Numrich lists the magazine model number of the Model 165 as having a “C”.
The Triple K magazine for the Model 165: