An article in The Hindu entitled "Itihasas come first" and published on 18 December 2017 (at a time when the Wikipedia claim that you're asking about didn't exist in the Wikipedia page) claims that:
An Itihasa is a record of something that happened, written by someone who was witness to the events. There are two Itihasas, Ramayana and Mahabharata, and in both these cases the authors were also characters in their chronicles. [...] When a report is written soon after an incident, it is likely to be more accurate, and the Itihasas have this advantage. That is why they have more authority than the Puranas.
But The Hindu is a newspaper, not necessarily a reliable source for definitions of literary terminology. What do more academic sources have to say?
From the article on Itihasa in Rachel Dwyer's Key Concepts in Modern Indian Studies (paywalled on the publisher's website and available on Google Books):
The literal meaning of the Sanskrit term itihasa (iti-hi-asa) is 'thus indeed it was' and it refers to legend, traditional narratives, events from the past and particularly those connected with past heroes. By itself, iti is often used to indicate preceding words or to suggest indirect speech or refer to what is known. It refers to that which has gone before, and in this sense itihasa since the nineteenth century has been used for the discipline of history.
Associated with itihasa are a number of other words. The aitihasika is the one who knows about the past and is familiar with traditional and historical legends. The most frequent association of itihasa is with purana which literally means 'belonging to times past' and to that extent is thought to encapsulate traditional history. The difference between the two terms is difficult to gauge and they are often linked as a compound term. The Puranas took the form of texts originally recited by the suta ('bard'), and later by the Brahmin priests. As a narrative recounting of the past, itihasa is also sometimes linked to anushasana ('governance'), and to dandaniti ('administration of justice') (Arthashastra 1.5.14).
[...] Itihasa was taken out of its ritual context and was now seen as central to governance. It took the form of charitas ('royal biographies'), prashastis ('eulogies') or histories of dynasties recorded in inscriptions, and vamshavalis ('chronicles'). These texts, recognised as itihasa, were akin to a more secular historical tradition and constitute the major sources for reconstructing the history of this period. History, as we know it today, is a post-Enlightenment concept where the narrative of the past is based on factual evidence. Itihasa, as used in premodern texts, has a wider meaning to include historical consciousness and perceptions.
Further reading: Bhattacharya 2010; Singh 2003; Thapar 1986.
-- Romila Thapar, Professor Emerita of History, Jawaharlal Nehru University, New Delhi
From the article Ranjan Ghosh, "Rabindranath and Rabindranath Tagore: Home, World, History", History and Theory (2015):
Itihasa as a narration or tale of human progress has its own worldviews (encompassing both the nonteleological and the domain of political action), an intelligible understanding of situations, events, emotions, and development without a kind of inscrutable mystique that prejudicially becomes attached to this word. Absolutes cohabit the synchronic; the defined and directed are simultaneous with the unpredictable and serendipitous. Itihasa for Tagore creates a delicate interpenetration. [...]
Itihasa mediates with our cultural past as much as it participates with the world outside our culture and tradition. So the point from which history is required to be viewed stands questioned: is it the point that world history formulates? Should it be the optimum methodological point that rationalizes the clash of historical positions or the organic point that views historical experience as an intricate mix of rhetoric, emotion, affective viewing, and intelligibility? The debate, perhaps, centers on how “ownness” and itihasa are connected. For Tagore, Katha or Jatra demonstrates their owning of itihasa by working out this delicate problematic of historical distance: narrative time, context, rhetoric, the sense of fact and fiction coming into a compound play. Jatra demonstrates the repetition that history generates: the same content being presented in a variety of retellings, resulting in different forms of historical affect, alienation, and kinaesthesia. Itihasa comes with a connotation of yatra, the journey, produced through pramanas (means of evidence/knowledge), perception, testimony, and inference in both the past and the present. To this, yatra, with its own means of knowledge-production, Tagore added memory (smriti). Itihasa is both presentative and representative—a racanashalay, a room for creation where pramanas come both veridically and affectively. Distance and desire in itihasa synergize to build their own creativity (srishtikartritva).
Neither of these detailed academic texts about itihasa mention any requirement that the author or narrator must be an eyewitness to events or part of the story. Therefore, I conclude that this is not a strict requirement and part of the formal definition of itihasa. I don't know where The Hindu got this claim from, but it could easily be that the Wikipedia claim originally came from The Hindu.