I think the received version in the West comes from a rather biased source. For example a recent book says
Allegedly, the Greek Supreme Court of Appeals, in an individual case concerning public liability for wrongful acts of Greek military in Cyprus, also found that Turkey intervened on 20 July 1974 “due to reasonable ground according to the Treaty of Guarantee”.
The fact that they put "allegedly" in there suggests that the decision is hard to verify. The fact is indirectly cited via Z. Necatigil, United Nations resolutions on Cyprus, in: Resat Arim (ed.) Cyprus and international law, p. 60 (67)
A bit more search finds that
Zaim M. Necatigil, formerly the Attorney-General of the TRNC, is a lawyer and legal consultant at the Ministry of Foreign Affairs and Defence, Turkish Republic of Northern Cyprus
So this is not really an unbiased source for material on Cyprus. But Necatigil has published a lot of material in the West, including a book (not that one) by Oxford University Press (also on Cyprus). So he is/was probably influential in some Western circles. The Turkish lobby in the US used to be well-known for its influence.
According to ProPublica, Turkey is one of the top foreign lobbying interests in Washington, ranking in the top list with the United Arab Emirates, Saudi Arabia and others.
Through various contracts and subcontracts, the roster of those officially lobbying on behalf of the Turkish government has been impressive: former CIA Director Porter Goss, Dick Gephardt, former Republican congressman Jim McCrery, former Democratic Senate Majority Leader Tom Daschle, a firm that employed former Trump campaign manager Paul Manafort, former Minnesota Republican congressman John Vin Weber, and more. And of course, perhaps the most high-profile fighter of Turkish interests in DC was former National Security Advisor Michael Flynn, who retroactively registered as an agent of Turkey.
Dan Burton is not mentioned there, but who knows from where he got that info.
The Turkish Coalition of America does have a 2007 article praising Burton's "history lesson" to Congress about Cyprus.
Pro Publica does list Burton more generally as having taken FARA money, but Turkey is not explicitly mentioned among the donors, which are not detailed there:
Spotlight: Who gave
Of the 74 donations FARA agents made to members they’d personally contacted, 34 came from lobbyists working for a single firm, the PLM Group. PLM was formed in March 2007 as a joint venture of the Livingston Group and the Podesta Group. In all, lobbyists with the PLM Group reported giving nearly $63,000 to members of Congress. Three of the contributions made to Reps. Dana Rohrabacher, R-Calif., Dan Burton, R-Ind., and Donald M. Payne, D-N.J., came on the same day they met with lobbyists.
One would have to figure out who the middlemen (PLM group) channeled money for.
And the Atlantic reported on Burton's alleged benefits from such lobying, albeit not from Turkey but from Bahrain
The Bahrain American Council then paid nearly $21,000 to fly Burton and his wife to Bahrain in 2012. The investment paid off almost immediately. Burton returned to Congress to deliver a speech hailing Bahrain as "one of our most important allies" in the Gulf region. And he suggested the antigovernment protesters there, who had been violently squelched, might "have been infiltrated by outside radical elements supported by Iran."
A much older article from 2001 in The American Prospect made more explicit accusations tying Burton to the Turkish lobby:
Zaire was not Burton's only overseas interest. He also took thousands of dollars in legal contributions from people with business interests in Turkey. Afterwards, he made a statement on the floor of the House of Representatives he had cribbed from a Turkish government official. The statement defended Turkey against well-documented charges that its government committed serious human rights violations against the Kurds.
In 1996, Burton made another floor statement that almost exactly echoed materials that Turkey's lobbying firm gave to members of Congress, according to The Los Angeles Times.
Anyway, I think all-in-all there is good reason to doubt Burton's "history lesson" on Cyprus.