Is that, then, truly a "free night"?

Most would say yes, since they're not planning to cause damages or use room service or minibar, and so they don't expect to be charged anything as a result of their "free night".

Some would say no, since they're planning to throw a raging party and trash the place, and don't want any risk of being charged money for that outcome. The credit card in both of these scenarios is used for risk prevention, not for revenue.

"Free tier" is a statement of how much you can expect to be charged if you stay within its boundaries, not a statement of your right to participate without accepting the risk of financial penalty if you choose to act outside those boundaries.

(A few would say no, since the principle of "free" means that any conversation about "money" is inappropriate — no matter how much risk that places on others. Other reasons surely exist too.)

It's still free because you aren't charged for usage within the free tier limits.