What is the authority as to why we do so?'
A: I don't know. I probably picked it up from a form on the first brief I ever did and just kept doing it that way. The only rule I know of is FRAP 32(a) Form of Briefs which reads in pertinent part:
The cover of the appellant’s brief must be blue; the
appellee’s, red; an intervenor’s or amicus curiae’s, green;
reply brief, gray and any supplemental, tan. The front
cover of a brief must contain:
(A) the number of the case centered at the top;
(B) the name of the court;
(C) the title of the case;
(D) the nature of the proceeding and the name of the
court, agency, or board below;
(E) the title of the brief, identifying the party or parties for
whom the brief is filed; and
(F) the name, office address, and telephone number of
counsel representing the party for whom the brief is
Technically, you could probably restrict it to the role you play on appeal (Appellee or Appellant), completely leaving out the role you played below, and give adequate information. Still, it seems to be the standard to recite the party's position in the Court below. See e.g.
http://www.recordpress.com/pdf/USCA%208th%20cir.pdf (see sample cover).
There may be a rule. There may not be. Maybe there's a case. Maybe there's an en banc case (8 to 7) with a vigorous dissent holding that you should not identify the role of the litigant in the court below on the cover because it's a spoiler. Maybe there's a split between the circuits that for some incomprehensible reason the Supreme Court has not seen fit to resolve. I'm not going to comb the rules and case law to see if I can find it. Nobody in St. Louis is going to pitch your brief because you put too much information on the cover.