1. I have to write a letter. (The obligation is the result of another’s direct command or direction.)
— not necessarily. I really do have to write that letter to Visser this weekend – the one that points out all the flaws in the "another's direct command" argument.
2. I have a letter to write. (The obligation does not result from the command of another, but rather is a self-imposed task: it means ‘to be burdened with’ or ‘to feel it incumbent on oneself’.)
— not necessarily. Looking at my inbox now, for instance, I see that I have several posts to answer. It isn't a task; it isn't a burden; and I certainly don't feel it "incumbent" upon myself to reply.
Indeed, it may well turn out to be a distinct pleasure...
MrP