Prince's 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's Terrace. [2]
The location pin in above map is put near its east entry, where existing numbering begins.
The area Prince's 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's 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's Terrace for your perusal :
sources
1. Government Gazette, 28 Dec 1923 (link)
2. Old Hong Kong by Colonial (1933)
Comments
Caine Road : a district with many schools
By the 70s, Caine Road had been a neighbourhood of many schools established. In fact, this may have a longer history, when schools are smaller in size.
In education dept report of 1925, a vernacular normal school for women at 7 Prince Terrace is mentioned, with average attendance 39 then. [3]
Same level as Prince's Terrace and across Shelley Street, there was a Chinese middle school at 2 Leung Fai Terrace c.1926. [4]
Wah Yan College once expanded to include location on 33 Mosque Junction, before 1921.
These are just few examples around this one block in the 1920s. [5]
sources and notes
3. All 8 students in the Second Year Class were successful in the Final Exam.
In fact it is for training women vernacular teachers. This is likely the first HK teachers training school, only one of its kind, for women,
originally housed in Belilios Public School. Students number grew steadily to over 100 in few years. It removed to Lee Garden Street by 1927 end.
The one for men was also started on Hollywood Road c.1921, near Man Mo Temple.
refer : Gov't Administrative Report 1925, 1920, 1926, 1927; Chinese Mail (華字日報), 1927-1-1.
4. ran by Chung Hwa Education Society; e.g. Kung Sheung Daily News, 1927-7-19
5. one more example is Shun Sau Middle School (信修中學) for girls, founded in 1922 on 61 Caine Road; one aunt of mine had studied there