Prince Terrace is likely named after Prince Edward, who visited HK in 1922. It was formally named by end of 1923, same time as Lee Tung Street in Wanchai.
The street's area should cover more than the Douglas Villa as the Gazette mentions south-west boundary of IL 152 in Peel Street [1]
If as Old Hong Kong, there are two 'terraces' c.1933 then, one on Caine Road and one behind which is Prince Terrace. [2]
The location pin in above map is put near its east entry, where existing numbering begins.
The area Prince Terrace covered is quite large, actually back to back with Rednaxela Terrace, which is only a narrow path today (similar as in 1890s, I think).
Below photo is actually Prince Terrace, not Rednaxela Terrace as what the title says. (David, please help to update it and the 'Place', when free)
Embed the street view of 13 Prince Terrace for your perusal :
sources
1. Government Gazette, 28 Dec 1923 (link)
2. Old Hong Kong by Colonial (1933)