The only exhibitions that count are those organised under the auspices of the Bureau International des Expositions. These are the enormous ones typically named like "Expo (year) (city)". For example, Expo 2020 Dubai was six months long, had 24 million visitors, and covered 438 hectares. If the event is not one of these, then the applicant does not get the benefit of the grace period.
The grace period also only applies to disclosures at the event, or in consequence of it (e.g. media coverage from journalists who saw the invention). It does not mean that the inventor has carte blanche to display the thing at other events, show it off on TV, create a public website about it, etc. So the language in the question about "get the patent waiver kicking" is not quite right. A professional legal advisor could say more about ways to go public.
The Patent Rules cover, in Rule 5 and Rule 67, some procedural matters in relation to disclosure at exhibitions. These essentially answer the subsidiary question about "how does one indicate to the patent officer that the technology presented was indeed mine?" - the organizer of the exhibition will certify that whatever was displayed, is indeed the invention for which the patent is sought.
For the inventor to "display" the invention for the purposes of the grace period, it is not required for the inventor to be physically present at the event, pointing at interesting features of the invention. The point is that the public has seen the thing. If the inventor is asking somebody else to explain the technology on their behalf, then it would be prudent to consult an IP lawyer about the terms. Hopefully, there is one in the picture anyway.
The question refers to the novelty rules in the Patents Act 1977, s. 2. An invention's novelty is assessed against "the state of the art" as of the time a patent application is made. If material has been put into the public domain, then it forms part of the state of the art. So normally, exhibiting an invention in public would compromise whether it can be considered "novel" later on. But one of the exceptions is a grace period if that takes place no more than six months before making the application, for certain qualifying events only. The wording in 2(4)(c) is that:
the disclosure was due to, or made in consequence of the inventor displaying the invention at an international exhibition and the applicant states, on filing the application, that the invention has been so displayed and also, within the prescribed period, files written evidence in support of the statement complying with any prescribed conditions.
So what is an "international exhibition"? Well, the 1977 Act was made in part because the UK had signed the European Patent Convention in 1973. That treaty says in Article 55 that a disclosure of an invention is to be discounted, within a six-month grace period, if:
the applicant or his legal predecessor has displayed the invention at an official, or officially recognised, international exhibition falling within the terms of the Convention on international exhibitions signed at Paris on 22 November 1928 and last revised on 30 November 1972
That is reflected in domestic law by section 130 of the 1977 Act, which defines "international exhibition" in just the same way. The IPO Manual of Patent Practice confirms (p74) that:
Only an exhibition which falls within the terms of the 1928 Convention on
International Exhibitions (as modified by the 1951 Protocol) is regarded as an international
exhibition. Such an exhibition has to satisfy stringent conditions; for example, it must run for
at least three weeks, and invitations to participate must be issued at government level
through diplomatic channels. A statement published in the Patents Journal that an exhibition
constitutes an international exhibition within the meaning of the Convention is conclusive
evidence of that fact. Regularly held events and trade fairs organised by particular
industries are unlikely to qualify.
Why just these events?
There is a long connection between these expos and IP law. The Vienna World's Fair of 1873 had some delegates refuse to exhibit their inventions, because they were worried that they would lose patent rights under Austro-Hungarian law at the time. When the Paris Convention was agreed in 1883, it included an article according temporary protection for inventions shown at these expositions. Because of the prestige of expos and the existing international structure around them, the concept of protection has been carried forward through the 1973 treaty and to the present day.