Hello,
so it seems the Hifiman is often recommended as a more "high fidelity" or quality headphone over those other two. Is this correct?
And if so, would you say, it has to be specific model (S or I) for that?
It's not correct, they are similar in the big scheme of things.
You're preferences will play the bigger role at this point.
I've preferred the K712 over the HD800 multiple times in the past. I've owned them both side by side for ore than a year. Other critical listeners I know also preferred the K712 over HD800 multiple times.
Recordings play an important role, you can not focus only on the headphone's frequency response and forget about the recording's tonal balance.
I'm not saying K712 is better than HD800, but it's good enough to please very critical listeners, sometimes more than the HD800.
You can see graphs or ask here which is best, which is higher 'tier', but the very nature of the sound of a headphone is very complex to describe as it involves many variables. It also depends on your setup, your recordings, your listening levels, your ears, you know... If you want a simple answer to your question, HE400i/s and K712 belong to the same 'tier'.
If you want to get the right headphone or at least try to gain experience in the game, then 'tiers' won't help you. Focus on what you enjoy and what you don't enjoy in what you have. Share that with the people here. Then listen to other headphones and ask for
comparative impressions, make sure to share a few of your recordings. For example, if you normally listen to Adele, and you read the impressions from someone who normally listens to live opera, those impressions won't serve you (this is key).