There can be benefits to both depending on where you’re working in relation to your FTP.
Generally speaking, I’d say that power is probably the better option in terms of defining training zones.
If you want to look specifically at effort, HR can be a good measure though.
Ultimately, unless you’re regularly lab testing fat/carb oxidation and lactate alongside your power/HR on the bike, these are only going to be estimations that won’t always be 100% accurate but are good enough for most of our training purposes.
The goal when setting these zones is to find your lactate threshold (it’s the easiest to estimate) and then base all of your zones (HR and power) on that one figure using a standardized formula. This means that your zones (again, both HR and power) might not always align 100% with reality.
For instance, your LT1 might be at a different percentage of your LT2 than mine, but we’re still using the same formula to calculate our zones. This means that your LT1 might be at the high end of zone 2 or maybe even the very bottom of zone 3 whereas mine might be at the middle of zone 2. Again, without lab testing it’s impossible to know.
Something I’ve done in the past is try to keep an updated and accurate FTP and lactate threshold HR so that I can look at those two data points together after my workouts. If I find that during certain efforts my zones are slightly off, I could probably assume that a good target for training purposes would be right in the middle of the two. From there I use power data since it’s the most stable, and consistent.
Your example of lactate/threshold training applies here. If your HR is in the threshold zone way before your power gets there, there is likely an issue with either your HR zone calculations or your FTP. Comparing these two data points helps in those situations. You should feel somewhat on your limit when riding at your FTP for longer durations of time. Sweet Spot is also hard, and it can be tricky to differentiate them sometimes, which is normal, but your HR should be relatively steady during both types of efforts.
Regarding higher power zones, I’d also recommend utilizing power data as HR lags far too much and isn’t always as steady during these harder efforts (it tends to drift upwards throughout the effort). In those cases, start your interval when your power is where it needs to be. Your HR will catch up and the effort is what matters in those cases, not your body’s response quite as much.
How are you calculating your HR zones? A common way that people do this is by using max HR figures which are actually really hard to accurately obtain and the formula is not super accurate in my opinion. I’d recommend doing a LTHR test if you haven’t to find and calculate your zones. It’s been more accurate for me at least.
Let me know if this helps!