What did he lie about?
In my not so humble opinion, the most critical issue about which Ted Cruz is being dishonest is that he is not, has never been, and never will be, eligible to serve as president of the USA, and by extension, also the vice-presidency.
As a reportedly exceptional student of constitutional law, and as an exceptional debate competitor, his attempt to usurp the highest office in the country cannot be a simple oversight or misinterpretation. His campaign is a deliberate attempt to undermine the US Constitution.
Therefore, any claims Ted Cruz puts forth that he is representing the best interests of US citizens in any manner is a blatant lie of the worst kind.
What did he [the author] lie about?
The pro-Cruz author lied about Trump having been a Democrat seven months prior, or at anytime in the past. Just like Cruz, his sycophants just make stuff up.
Trump has been a Republican since at least the days of Ronald Reagan's presidency. He did register with the Reform Party for a brief period to run as a Reform Party candidate many years ago. Trump has said that as a businessman, he has had to contribute to politicians of both parties. I've seen no evidence that Trump has ever been registered as a Democrat.