It's not clearly answered, and the possibility of branching timelines exists (which makes sense, given RAH's many-universes concept), but it seems that he knew what was going on before he went back in time, because he saw in the register at the County Clerk's office in Yuma that He and Ricky had just gotten married, and leapt to the right conclusion. (His next actions show he was thinking of the time-travel aspect; acquiring gold and wearing it under his clothing, reaching out to Dr. Twitchell, etc.)
Then, just to make sure to keep the loop closed, he:
hired another cab and we jumped to Yuma. There I signed the county clerk's book in a fine round hand, using my full name "Daniel Boone Davis," so that there could be no possible doubt as to which D. B. Davis had designed this magnum opus.
He does ask the same question you ask, tho:
Why didn't I see the notice of my own withdrawal? I mean the second one, in April 2001, not the one in December 2000. 1 should have; I was there and I used to check those lists. I was awakened (second time) on Friday, 27 April, 2001; it should have been in next morning's Times. But I did not see it. I've looked it up since and there it is: "D. B. Davis," in the Times for Saturday, 28 April, 2001.
However RAH doesn't lock off the possibility that you are asking about:
Philosophically, just one line of ink can make a different universe as surely as having the continent of Europe missing. Is the old "branching time streams" and "multiple universes" notion correct? Did I bounce into a different universe, different because I had monkeyed with the setup?
But, although RAH doesn't say for sure, Danny has this to say:
I probably felt asleep that night and missed reading my own name, then stuffed the paper down the chute next morning, thinking I had finished with it. I am absentminded, particularly when I'm thinking about a job.
But what would I have done if I had seen it? Gone there, met myself-and gone stark mad? No, for if I had seen it, I wouldn't have done the things I did afterward-"afterward" for me-which led up to it. Therefore it could never have happened that way. The control is a negative feedback type, with a built-in "fail safe," because the very existence of that line of print depended on my not seeing it; the apparent possibility that I might have seen it is one of the excluded "not possibles" of the basic circuit design.
So, like a number of other RAH time-travel stories, this has some chicken/egg aspects, but the above quote suggests that there has not been any actual change, as the change could have precluded itself. (Bootstrap paradox.)
Then he, like other RAH characters, drops the issue as the philosophical aspect isn't terribly important to someone who has found happiness.
But I don't worry about philosophy any more than Pete does. Whatever the truth about this world, I like it. I've found my Door into Summer and I would not time-travel again for fear of getting off at the wrong station.
So, in the end, it's not said, but the main character really doesn't care.