Your first summary is the historically accurate one. You could add:
First off, the heliocentric model had been proposed by Copernicus nearly a century earlier, and without controversy. It is estimated that roughly half of the clergy (who also happened to be the academic community at the time) already believed that the earth orbited the sun. What set Galileo's work apart was that he claimed to have proved the heliocentric model using the tides as evidence. That is the part with which the Pope disagreed. Of course, we now know that Galileo was wrong about his proof. It wasn't until advanced telescopes were invented in 1830 that science was actually able to prove the heliocentric model.
The Pope was also Galileo's patron, and was sponsoring the book. And when Galileo published the book, he openly mocked the Pope in it. The Pope, of course, overreacted in a big way, placing Galileo under house arrest and banning the book.
At the time when Galileo's book was published, the Roman Catholic Church officially maintained that scientific proofs outweighed Christian doctrine (on the grounds that doctrine is more likely to be misunderstood). Science was granted an equal footing with doctrine, and statues of Plato and Aristotle (as the founders of western science) were added to the statues of the saints on the walls of many cathedrals. According to the Catholic view, all truth was God's truth.
Contrary to popular opinion, the Church, at that time, did not attempt to suppress dissenting views. In fact, in order to earn a doctorate degree, candidates were frequently required to construct a thesis against the accepted view.