Even today, in many jurisdictions, it is legal to go by any name you choose provided the purpose is not to deceive or to obscure another crime. The procedures to "legally change" are mostly to do with the convenience of having your new name known and acknowledged by other people you deal with (on your bank account, drivers licence, passport etc). As late as the nineteenth century, that second stage of a change was not a concern. The people who needed to know your name probably met you everyday or at least weekly at church or the market. So it was quite possible to simply choose a new name and use it.
As Ezri indicates, sometimes the change was made by another person charged with recording the name. An illiterate man was unable to tell that the spelling had altered since last his name was written, he just made his mark beside the words.
Two related issues for you as a family historian are (a) was the name stable before 2xgreat grandfather (What was 3xgreat grandfather called?) and (b) how you record and document the change(s).