There is a pedigree of the family of Errington of Errington in at
least two published editions of the visitations of Northumberland (the
editions by marshall and by Foster, the first of which is available
online at the Internet Archive). There is also a pedigree of the same
family, covering further forward than the visitation pedigree, in vol.
4 p. 188 of the History of Northumberland, also available at the
Internet Archive. Where the pedigrees in these sources overlap, they
match closely if not exactly.
In addition to Errington of Errington, there were other Errington
families in Northumberland, notably in Ponteland and Denton.
Pedigrees of these last two families are in the visitations and also
in vols. 12 and 13 of the History of Northumberland (which are not
available online AFAIK). The Ponteland and Denton families are
closely connected to one another, but I don't know of a documented
connection between them and the family of Errington of Errington.
The String of marriages you mention above (Horsley, Creswell, et al.)
appear in the pedigrees of Errington of Ponteland and Denton but not
those of Errington of Errington. OTOH the marriage of John Errington
and Dorothy Widdrington does appear in the pedigrees of Errington of
Errington. However, in the History of Northumberland, this John
Errington is noted as having died by 1626. So he clearly cannot have
had a daughter Isabel b. 1688 who married James Charlton b. 1685.
I'd guess that your undocumented pedigree has mashed together segments
of different Errington families (and there are more families than just
the three mentioned - see the visitation pedigrees and especially the
noted volumes of the History of Northumberland).