Mr Cessna,
I'm surprised that the A/G person at Seething made such a fuss of it. However, it isn't "normal" for two aircraft to talk to each other directly on an airfields frequency when the ground radio is manned and operating.
My suggestion in a situation like this is to make a call to the ground station along the lines of
G-XXXX is at (your location). Not visual with the reported traffic, say again his position.
That is then an open invitation for the other aircraft to report his position to the A/G radio unless the A/G replies.
Then you can't be accused of talking directly to the other aircraft but you have achieved your aim.
I do think that radio operators who spend time on the radio telling pilots off are unprofessional. If the pilot is a student, then they are just overloading him/her and a word later on the ground is much better. If it is a qualified pilot then a word afterwards is also much better.
Duxford once spent an age giving me a bollocking on the radio insisting that I did not have PPR. Wasted minutes where no one else could speak.... and they were wrong, they had mislaid the piece of paper. As an inbound pilot all I could do was argue with them while circling or go home.