The legal process between creating a public right of way and a private right of way is similar but the latter involves more people (there is no specific number).
It can arise after unchallenged use by the public for at least 20 years, which may then give rise to a presumption of dedication under Section 31 of the Highways Act 1980.
Or it may arise under common law after any appropriate period of time (again not specified) by way of a presumed deed that has been lost (known as a "modern lost grant").
This isn't what happened here, and only that individual is legally entitled to the private right of way - thus it technically excludes the other two houses whose fence-line has also been changed.