In the clink

I had no luck searching for the origin of "getting thrown in the clink" on your website. I did find the following on the taboo website whose origin (URL) I will not mention here:

The name of a prison which was on Clink Street in the Southwark area of London.
Claudie wrote: I always thought that the doors of the cells went "clink" when they shut behind the prisoner. Maybe a more accurate name would be the "clang".

I had heard the second (clink/clang) meaning and origin but not the first. Do you know what the origin is?

This is what I believe is the origin:
The "clink" is the name of a specific London prison which, in turn, took its name from the Borough in which it was sited. This was The Liberty of Clink, a district of Southwark exempt from the jurisdiction of the City of London.