Yes, you do have a .32 Safety Hammerless First Model revolver. However, since the cost of a letter is at least a third of the value of the letter and unlikely to increase the value of your revolver, I wouldn’t necessarily purchase one.
I disagree with the date of manufacture of your revolver, or at least the frame thereof. As your revolver is considered to be ANTIQUE (which is the status of all of these manufactured up to serial number 91400), the frame was manufactured prior to 1898. In fact, as frames were manufactured in batches, and the First Model was up until serial number 91417, I think antique status should include all examples up to 91417, but a reasonable arbitrary number was selected (I can guarantee you there was not a post 1898 batch of 17 manufactured). Now, your revolver may have been assembled as late as 1902 and the ship date is known, obtained through a letter. But, unfortunately, these aren’t worth that much making a letter impractical for all but the very best examples.