I was told in a lecture that no two (true) democratic states have ever gone to war. How true is this, and what examples are there?

It's largely a definitional question. As long as you can define democracy tightly enough you're golden.

All in all there are probably two to three dozen decent candidates for democracies going to war with each other. In each case you have to have a discussion over what counts as a "True Democracy."

Does Fujimori's declaration of Martial Law make it not a democracy for the Equadoran-Peruvian border war of 1995? Does president Lincoln's suspension of Habeas Corpus mean the US wasn't a democracy in the Civil War?

How broad does enfranchisement have to be to be a "True Democracy?" Does the existence of slavery disqualify the US prior to the late 1860s. Does Jim Crow disqualify the US until the 1960s? Do similar issues disqualify the Boer Republics? Do women have to be enfranchised? If so does that mean Switzerland wasn't a "True Democracy" until the 1970s? Does the existence of a non-democratic element like a House of Lords disqualify a country? If not how far does that go? Was the U.K. truly democratic in 1914? If yes, was Germany? It had regular contested elections and the Reichstag had some real power. Even if the Kaiser held the bulk of power in the German system, he wasn't an autocratic ruler like the Tsar. So where is the dividing line between a democratic system with non-democratic elements and a non-democratic system with democratic elements?

How about civil wars or revolutions? Both sides of the US Civil War had democratically elected governments. Same with the US Revolutionary War. For a more recent example, the Croatian War of Independence in the early 1990s was fought between a democratically elected Croatian government and a democratically elected Yugoslavian government.

So on and so forth. Basically the only way you can argue that democracies don't go to war with one another is by adding in that "True" adjective. Then you can toss out any counter examples and declare the rule to be correct. Of course by limiting it to "True Democracies" you're also making it less significant. Under deep scrutiny, just about every democracy could be eliminated. Democracy in practice tends to be a bit messy and flawed.