My understanding and usage of the word "prosecution" is in the sense of "pursued by the regulatory authorities for breaking a law". In the same way wars are said to be prosecuted. This includes driving offences. I still believe I can be prosecuted by the police for a simple non criminal speeding offence. Correct me if I am wrong, IANAL but I don't agree with your definition.
The ASA decided in this case not to prosecute. I have been in a similar position where a company I worked for broke the food labelling laws. This can be prosecuted under the Food Safety Act, etc. Some of the terms of the FSA are criminal, I believe some not. Was an offence committed? I believe so. However we were not prosecuted because Trading Stds decided not to, they soft policed it and gave us a talking to, slapped us on the wrist, told us to scrap and reprint the labels, etc. Now in the letter of the law there was no offence committed, because there was no conviction as such and nobody was prosecuted. Innocent until proven otherwise, etc. However in the spirit of the law, was an offence committed? As I say, I believe so. Similarly, I was spoken to by a policeman some years ago for exceeding a 60mph speed limit. He did not instigate proceedings. However had I committed an offence? You and I both know that yes I had. Yet I'm innocent.
Now we can dance around arguing about angels on the head of a pin and semantics, or not. We can spend all day defining "prosecution", or not. However the fact remains that RA was dealt with by the regulatory authorities for making false claims in advertising. I thought that he had been prosecuted for this. It turns out not. He was instructed to take down the offending terms. I imagine that he was told to do this or face prosecution.
Tesco were never prosecuted for their part in the horsemeat scandal. No offence committed, then? Apparently not. Well, there hasn't been a conviction of Tesco, has there?