You are probably right to be skeptical of the image from the House of Names website, which appears to be an incorrect rendering of the arms of "Ashford, or Ayshford (Ayshford, co. Devon, and Cornwall; the last male heir, John Ayshford, Esq. d. in 1688; the heiress m. Sanford, ancestor of William Ayshford Sanford, Esq., of Nynehead, co. Somerset: a branch of the family settled at Wonwell, in Kingston, co. Devon, and is now represented by L.L. Ayshford Wise, Esq.)." (cited in Burke's "General Armory.")
The arms in Burke are blazoned "Argent between two chevrons sable three ashen keys azure." The image from HoN has the chevrons between the keys, and not the keys between the chevrons as given in Burke.
Bardsley's "Dictionary of English and Welsh Surnames" cites "Ashford, Ashforth, Ayshford - Local, 'of Ashford,' parishes in diocs. of Canterbury, Exeter, Longon, Southwell, and Hereford; v. Ford and Forth for the two suffixes.
John de Esseford, co. Oxf., 1273. A. Eudo de Assheford, co. Linc., 20 Edw. I
1563. buried - William Ashforde, prentis to Roger Beawe: St. Mary Aldermary, p. 135.
1594. Thomas Aysheford, co. Devon: Reg. Univ. Oxf. vol. ii, pt. ii, p. 208.
1773. Married - Joseph Ashford and Charlotte Probert: St. Geo. Han. Sq. I, 228."
So the names Ashforth and Ashford are clearly related, but that still doesn't mean that any Ashforth families would have the right to use the arms of the (now-extinct) Ashford line.
Burke's "General Armory" cites four other Ashford coats of arms, two variants on the white shield three items (bunches of keys, pineapples, escallops) between two black chevrons; one with a chevron between two other narrow ones, the large central one charged with three pineapples, and the other one a green chevron between three green keys. Burke lists no Ashforth arms.
I hope that this information is helpful to you.
David