The difficulty is recalculated about every 2 weeks, the hash rate is estimated every block or so, then they use those values to estimate the halving date. When blocks get solved faster more often, the estimated date moves closer or earlier.

As of this post, the ETA is back to May 2024. I wouldn't get too excited about the exact date as we won't really know for sure until about a few months before it is supposed to happen. It may happen earlier or later, usually earlier, but more than 4 months earlier is kind of stretching it. The bitcoin network will recalculate difficulty to adjust for increased hashrate, so estimates might move in 2 weeks, then it will try to normalize. The target will always be as close to May 2024, but nevermind the date or time, it's the block height.

All these count down websites still calculate based on the block height.

It's like saying, wait for 1 block or 1 confirmation, but that does not necessarily mean wait for 10 minutes; most of the time it will be close to 10 minutes, but we all know it can be 1 minute or 1 hour before the next block. The same with halving block. It's still going to be May 2024. April or March 2024? Possibly. But 2023? Very unlikely. Not impossible, but not likely.