What is the formal definition of a stellar day? I'm having trouble understanding precisely what a stellar day is. Neither the USNO nor the IERS sites provide a definition. And Wikipedia's description as the "rotation period relative to the fixed stars" as "the span of time it takes for the Earth to make one entire rotation with respect to the celestial background or a distant star" is confusing, since, relative to fixed stars, the earth is both rotating and precessing, so that the rotation period relative to fixed stars as such, is not a single value, but will be different for stars with different equatorial coordinates. 
I assume that the stellar day is simply the earth's rotation period on its axis, but I'm not sure that's right, or how to state it formally (e.g. with respect to inertial frames).

I understand why the stellar day is distinct from and longer than the sidereal day, since the coordinate system that defines a sidereal day is rotating slowly against the rotation of the earth.
 A: According to the Wikipedia article you reference, "stellar day" is supposedly a new name for a planet's sidereal rotation period. However, I cannot find any documentation of this new name anywhere, and that includes my copy of Explanatory Supplement to the Astronomical Almanac, 3rd edition, edited by Urban and Seidelmann (University Science Books, 2012) which was just published within the month. This source is definitive and the term doesn't appear therein (okay at least not in the index). However, further digging found reference to it here
http://hpiers.obspm.fr/eop-pc/models/constants.html
but I've yet to find an actual statement of the change in terminology from "sidereal rotation period" to "stellar day" anywhere in the IERS conventions. Anywhere, the distinction is that the sidereal day is measured relative to the moving vernal equinox, which accounts for precession, whereas the sidereal rotation period (stellar day) is relative to the fixed inertial frame of background stars. 
A: The sidereal day is the (mean) time between two transits of the R.A. origin, i.e. the vernal equinox. You are right, and this is strictly not equal to the time it takes the Earth to do a rotation in relation to the fixed stars, because the Vernal Equinox itself is precessing (and nutating).
The amount of time it takes the Vernal Equinox to do a full rotation is suposed to be well known (26000 yr) so you can simply make the correction: since the Vernal Equinox moves towards west, that period you call "stellar day" must be slightly larger, and you can do the correction very easily (I don think there is a more formal definition based upon any reference star, but please post it here if you find it).
Just for curiosity, in which context are you using that "stellar day"? I had never heard about it. Astronomers use the sidereal day as a synonym. The difference is surely much smaller than the "imprecisions" of the Earth movement itself, so it should have had no sense in the past (nor today with atomic clocks).
(Not to be confused with the Solar Day, i.e. the time between two transits of the Mean Sun, which is the "normal" 24-hour day, about 4 minutes larger than the Sidereal Day)
A: I think the original post was asking what a stellar day is as the sidereal day is a rotation of the earth and a solar day is a sidereal day + the missing 3 minutes 56 sec needed to = 24 hours. So why do we need a stellar day and what is it used for? To my understanding Stellar day is the more accurate name of a full 26,000 years rotations as a sidereal day is shorter by 0.0084 seconds. 
Stellar day = the longer "true" sidereal period as note by the International Earth Rotation Reference systems Service
A: In the traditional way of locating stars, astronomers measured everything from the "first point of Aires", the R.A. origin. The time it took this point to back rotate around to the same point in the sky is the definition of a sidereal day. Since the earth's axis processes, this point moves in the sky each sidereal day. A newer way of measuring is to use the Earth Rotation Angle (ERA). The time that it take for the earth to rotate 360 degrees relative to disant "fixed" stars is called a stellar day.
