Regarding the Michael Carrington at the Siege of Acre story--there doesn't seem to be anything to confirm that he was there, much less standard-bearer.
For what it's worth, other families make the same claim.
"Lord Simon Manning was said to be the royal Standard Bearer to King Richard the Lionheart. He carried the royal Standard to Jerusalem in 1190 during the First Crusade."
http://wc.rootsweb.ancestry.com/cgi-bin/igm.cgi?op=GET&db=jwebe...
http://manningmusings.blogspot.com/2016/04/simon-de-manning.html
Wikipedia shows Peter (or Pierre) de Preaux as Richard's standard-bearer, but doesn't include a footnoted source to back that up.
"Peter and [his brother] William accompanied King Richard on crusade in 1190. Peter was assigned the dignity of royal standard-bearer over the hereditary standard-bearer of England, Robert Trussebut. Peter and William were known to be with the King at Vezelay, Marseille, Sicily, the conquest of Cyprus, the Siege of Acre, the Battle of Arsuf, the march to Jaffa, the advance on Jerusalem, Darum, and last but not least, the Battle of Jaffa."
https://en.wikipedia.org/wiki/Peter_de_Preaux
https://en.wikipedia.org/wiki/Standard_Bearer_of_England
Pierre de Preaux, (Prouz)
Under the name Peter de Pratellis, one can find many old references to his being standard-bearer.
One family tradition is that the armor-bearer of the early kings of England was of this family. Peter de Pratellis, the Crusader, was hereditary standard-bearer and so beloved of Richard that the king kept him always by his side. Peter de Pratellis and his brother William were as gallant and loyal knights as ever bore the cross. As plain Peter and William Pratt, however, their exploits would certainly lose some of their romantic interest.
http://clickamericana.com/eras/1900s/variants-of-the-surname-pratt-...
"John de Pratellis was a favorite minister of Richard Coeur de Lion, and he and his brother Peter, hereditary Standard Bearer, with the Archbishop of Canterbury...."
Daughters of the American Revolution Magazine, 1921
https://books.google.com/books?id=aO1RAQAAMAAJ&pg=PA580&lpg...
The Dragon-standard, of which we have seen some examples in our first division, is still found among the Germans and the English. We have already observed its exact form in the pictures of Harold in the Bayeux tapestry. It accompanied the hosts of Richard Cœur-de-Lion. Richard of Devizes, in recording the attack upon the "Griffones" at Messina, says: "The king of England proceeded in arms: the terrible standard of the Dragon is borne in front; while, behind the king, the sound of the trumpet excites the army[231]." Hoveden, under date 1191, tells us that Richard "delivered his Dragon (Draconem suum) to be borne by Peter de Pratellis."
Ancient Armour and Weapons in Europe: from the Iron Period of the Northern ...
By John Hewitt
http://ocw.nust.na/gutenberg/4/6/3/4/46342/46342-h/46342-h.htm
Richard L being in the Holy Land, " tradidit Draconem suum Petro de Pratellis ad portandum contra calumniam Rob. Trussebut," &c. Claus. 28 Henry III. n. 7.
https://books.google.com/books?id=7LIRAAAAYAAJ&pg=PA32&lpg=...
Certe inter Anglicorum Regum vexilla [] unum exstitisse, Draconem vocitatum, auctor est Rogerus Hovedenus ann. 1191. illudque a Ricardo I. et in ea, quam iniit contra Saracenos expeditione delatum :
Cum Rex Angliæ fixisset signum suum in medio, et tradidisset Draconem suum Petro de Pratellis ad portandum, contra calumniam Roberti Trussebut, qui illum portare calumniatus est de jure prædecessorum suorum, etc.
http://ducange.enc.sorbonne.fr/DRACO
So we can, at least, debunk the old "standard-bearer" chestnut!