I have found some weird work around to achieve what I need.
But it involves CB Conditionals with a weird conditional a Custom HTML field with substitution overriding the display and luck.
I believe this is too much to be handy.
It would be great to have a simple solution to assign to each field the registration step on which it will be displayed independent from the profile view tab.
A text field should get displayed on tab A for registration. It should show on tab B on profile and it should not display in user list.
So I have assigned the field to tab A and set a CB condition that is always true to hide the field on profile. Then I created a Custom HTML field with substition to show the field.
However it does not show.
While the avatar field is set to show on user list, the text field is not. I believe this is the reason why the substition does not work.