Keith
I have nor seen a Tiffin school cadet corps cap badge. Lots of photos of cadet units pre-WW1 show caps being worn without badges in them so you can have a cadet corps without a cap badge being made for it.
The existence of 2 different s/ts is not unusual as often cadet corps were raised by the schools and later on were affiliated to a local regiment so could have 2 s/t types. Once they were affiliated then they would wear the local regt's badge. In Tiffin's case this was 1915 according to Westlake and he records the s/t that you mention. They are in the 1916 Army List as a result. Page 1195
https://deriv.nls.uk/dcn23/1232/4822/123248228.23.pdf
In Tiffin's case the local regt in Kingston was the 6th Bn so that ties in nicely. Was this the bespoke 6th cross badge or the bb East Surrey Regt badge?
Alan