Intuitively, I think there are several audiences at play here, each of which is reading something different into Trump's threat. For context, here is what he said:

> Iran is talking very boldly about targeting certain USA assets as revenge for our ridding the world of their terrorist leader who had just killed an American, & badly wounded many others, not to mention all of the people he had killed over his lifetime, including recently....

> ....hundreds of Iranian protesters. He was already attacking our Embassy, and preparing for additional hits in other locations. Iran has been nothing but problems for many years. Let this serve as a WARNING that if Iran strikes any Americans, or American assets, we have.....

> ....targeted 52 Iranian sites (representing the 52 American hostages taken by Iran many years ago), some at a very high level & important to Iran & the Iranian culture, and those targets, and Iran itself, WILL BE HIT VERY FAST AND VERY HARD. The USA wants no more threats!

The first audience for this is obviously Trump himself. If I ask myself what he's trying to accomplish here, and I believe that he's trying to intimidate Iran into backing off, at least for the moment. He's telegraphing that there is a strike plan already prepared and that it's something that Iran really doesn't want to happen. In my opinion, Trump is very nuanced - almost machiavellian, in fact - in his overall strategies but is very straightforward in his communication. If he'd wanted Iran to believe that he would order the destruction of cultural sites, he was have said it plainly. The mention of "culture" is the last in a series of descriptors, not the central point.

Also consider that Iran is a theocracy. As a result, presumably any attack on the Iranian government would target religious sites implicitly: there is no clear dividing line between "the government of Iran" and "Islam in Iran". Given that the government of Iran is the second audience, it seems that what he's saying is that his threat applies to government leadership instead of solely military targets.

Trump's domestic opponents are another audience, and have different biases. They seem to expect Trump to act aggressively and in a way ignorant of both the status quo and international law. I get why they are interpreting "important to [...] the Iranian culture" as "Iranian cultural sites", but that strikes me as a particularly uncharitable reading. This creep in what was actually said versus what has been interpreted as having been meant continues; the article linked is from an art-centric site. It uses "heritage sites" in the headline, and "heritage sites" aren't mentioned at all in Trump's tweets. Further, the article quotes art directors, and then explicitly calls out Persepolis as have been "in all likelihood" "spared thanks to the backlash against Trump's threat." Persepolis. In fact, none of the heritage sites listed seem like they would have ever been on a strike list in the first place.

I acknowledge that I don't share the biases of the article's author, but it doesn't seem at all reasonable to assume that ancient ruins are what were referenced in Trump's tweets. On the other hand, if my reading of Trump's intentions are accurate it would be unwise for Iranian leadership to be holding their press conferences in the Masjed-e Jāme’ in Isfahan or using the Shushtar hydraulic system as a convenient place to store centrifuges. It's unlawful (and immoral) to target a heritage site because it's a heritage site, but if the enemy attempts to use the site as a shield that's another story entirely.