Anonymous“But that would be "necessity" and not "obligation", would it?”
No; there is no necessity for me to write to Visser; and there is no external command or direction. I say "I have to" because I want to write to Visser.
Anonymous “Not necessarily" allows "possibly", doesn't it?”
To falsify your hypothesis, we need only find one exception.
Anonymous “So for you, MrP, this is possible "I have several e-mails to answer, but I feel no obligation or pressure to do so.", is it?”
Even better:
1. I have several emails to answer; but I feel no obligation, necessity, pressure, or constraint (external or internal) to do so.
(It's called "Friday afternoon in any workplace".)
Kooyeen “And by the way I am afraid that distinction is not even universal in the UK.”
I'm not sure it's even a true distinction. Cf.
2. I must do my homework, before I go to bed.
— subjective: the speaker has imposed an obligation on himself.
3. I must do my homework before I go to bed.
— objective: the teacher has imposed the obligation on the schoolboy.
Every obligation is naturally a combination of subjective and objective requirements.
Anonymous “I'm told that where it does occur in the UK, it is seen as an Americanism, i.e. as an import, or as the speech of young people.”
I've often heard people express views on split infinitives, terminal prepositions, "between you and I", etc. But I've never witnessed a "have to"/"must" debate, except on an ESL forum; and I've never seen this use of "have to" described as an "Americanism" – except by a certain lactescent poster here (who delights in vexing our US members, for reasons that have never been entirely clear to me).
MrP