I can’t figure out which PRS model I have…

How did you figure that out? I went on the website and didn’t get anywhere…
Serial numbers are sequential so they don’t help with model identification. You can take a look in the pickup cavities and you should see a sticker with an alpha numeric code on it (it could also be on a sticker on the outside of one end of the case). Google “Modcat decoder” and enter the code. It will tell you all about the guitar. Woods, finish, model,etc.

As far as how he figured that out:
The first 2 is the year identifier. In this case, 2002
The next group of numbers is the serial number. It falls in line with the range of numbers used in 2002
The 10 in the upper right indicates that this is a “10 top” which is a grading of the maple top used. In 2002 you would have had four grades of top; Core, 10 Top, Artist and Private Stock.

For your pictures it becomes apparent that it is a Custom 22. 22 frets and a trem make it so, along with the 5 way rotary. IIRC, the only other 22 fret set neck model made in 2002 was the McCarty. That would have a stop tail, 3 way toggle and vintage style tuners
 
As others have said, the 2 and the serial number indicates 2002 production. The 10 indicates a 10 top. Beyond that, it's the way the neck is attached to the body (set neck models with maple tops are usually "Custom" or "McCarty" models), the number of frets (in this case, 22), the hardware (this has a tremolo bridge and modern sealed gear tuners, whereas a McCarty would have a stoptail bridge and vintage style stamped housings), and the controls (two hum bucking pickups, volume, tone, and rotary switch) that identify this as a Custom 22.

PRS didn't have a huge number of models back then, so it's a simple process of elimination.
 
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