Running around with a surcoat that displays a heraldic achievement is related to heraldry. Running around in red and white clothes isn't heraldry, even if your heraldic achievement uses red and silver as its colors. For one thing, you (or you and your family) are the only ones entitled to use your coat of arms. Anyone are allowed to wear white and red clothes.
Hans
Livery is covered under heraldry. See Fox-Davies,
a Complete Guide to Heraldry. He makes several interesting observations, including that white and silver can be heraldrically distinguished in English heraldry, that Skye and Asure are different in East European Heraldry, and that Skye is used sans proper in some Italian arms.
Note also, Hans, items blazoned "proper" or "natural" can't normally be on fields of the similar coloration, but exceptions HAVE been made for both silver on white and yellow on gold. Mostly German arms; I've seen documentation of such on the rolls of SCA armorial submissions; I've also seen Laurel King of Arms reject them forthwith if they are for anything other than armed/langued, and not accompanied by scribal notes to use separate use of gold and yellow.
The list of heraldic shades in ACGTH is Or/Yellow, Az, White*, Gules/Red, Azure/Blue, Vert/Green, Sable/Black, Murrey/Sanguine/BloodRed, Mur/dark-reddish-purple, Purpure/Purple, Tenne/orange-tawny, Brunatre/"a brown color", Skye/Blue-Celeste, Amaranth/Columbine, Carnation/Flesh, Iron Grey, Earth, Brown, water and orange.
The list for ENGLISH heraldry is shorter...
Or/Yellow (no distinction betwixt them), Argent*, Gules, Vert, Azure, Sable, Purpure, Murrey, and Tenne.
*ACGTH, p. 70-71, noting that the royal label for the Prince of Wales is white, and is placed upon the supporters and crest as part of the achievement... including the argent unicorn.
I'll note that, in practice, the lion in the English arms ALSO bears a crown, and that the crown is of gilt, while the lion is often depicted yellow, if not tawny-yellow, despite both being blazoned Or. This is not, however, requisite.
I'll note as well that the current displayed Royal Labels are depicted on a white unicorn and are finely fimbriated sable despite not being blazoned such.