Since you are new to ancestry, I might point out a couple of things.
First, there WERE official adoption procedures in "those days" if you are talking about the US. You might find some valuable family information by looking for the adoption proceedings in the appropriate court of jurisdiction.
That fact notwithstanding, before birth certificates became our "modus operandi" in the US - generally after 1900 to 1920ish, depending on the state, people very well could change names and call themselves whatever they wanted in records that we might find for them.
When the name in a branch of, say Smith changes to Smythe in a line, I try to pinpoint the adult who first used the new spelling / name; which in that case is your grandfather. I will then use that name as my primary name for him or her. I have encountered this situation in numerous parts of my Hallock surname study, where various branches changed their spelling to Hallack or Halleck.
The more important question is who have you have designated as his PREFERRED parent, and whether you record the adoption at all. The PREFERRED parent is important when you import this file into Family Treemaker because if you designate the non-blood parent as a PREFERRED parent, it can effect who is include in ancestral reports and export datasets (a filter for exports is whether to include preferred parents or all parents.)
In your situation, I would go a step further and not record your grandfather Harry Bailey, as a child of Arthur Harry Bailey. This is because that action will cause your grandfather and all of his descendants to appear as descendants in descendant charts and reports of Arthur Harry Bailey, when there is no blood tie. I prefer to show only blood ties in my reports and in my file, but some people like to show both - which FTM will accomodate. You should at least consider the consequences of that decision before you make it.