Most email systems do not signal to others that you are online, and such signaling is not part of the email protocols themselves.
We usually say a lot of bad things about email but it’s a rare case of a technology which is independent, interoperable and if you use it right, can be non-distracting.
I really miss the old days of communication, it was such a simple system: when I was online (on whatever chat service) you could ping me, otherwise, you were able to send me an email. Nowadays we just install multiple messaging apps, each of them is constantly online. We receive multiple notifications and we try to fight the distraction with hacks like Do Not Disturb and AI that tunes our notifications.
Sometimes it works, but we should also train some basic expectations on response time to our peers: send me an email, or if it’s urgent, message me, but never expect an instant response.