The balled (headed) eagle?

Now I observe you staring upward and puzzling your wits to guess what great bird it is you see wheeling aloft over our heads. That, Sir, is the type, symbol, and adopted emblem of our nation, the BALD EAGLE, who, by the way, is not bald, any more than that stout and sturdy youth, the thriving republic, whose character he represents; only his head and neck are white; so is his tail; the rest of his body is brown. — New England Magazine, Vol 1, 1831

[emphasis in bold mine]

The Latin name for this iconic raptor is Haliaeetus leucocephalus, which means white-headed sea eagle. The Hali is Greek for “sea” and Aiētēs is “eagle”. Meanwhile, its species name is derived from leucos, “white” and cephalus which means “head”.

In 1782 the bald eagle was designated by Congress to represent the, then fledgling, United States. It is a fine and majestic-looking predatory bird but its head is amply covered with snow-white feathers: to put it bluntly, it is not bald. On the other hand, the immature bald eagle has a mixture of brown and white feathers until its fourth or fifth year of life when the distinctive solid-white head and white tail fully appear.

Etymonline says bald, meaning the lack of hair, is from the 14th century but its origin is uncertain, it suggests that it could have been derived from Celtic bal meaning “white patch, blaze”.
A Middle English Dictionary (Oxford, 1891) suggests that bald comes from the Old English for ball: beal, beall, dating back to c. 1200.

Etymonline also notes the following:

Bald eagle first attested 1680s; so called for its white head.

I also read that the raptor's name was taken from piebald (source), a word that first appeared in 1590 according to Merriam-Webster and means patches of black and white. Oxford Dictionaries states that pie recalls the magpie's black-and-white plumage, and bald (in the obsolete sense ‘streaked with white’).

So now I am quite confused. What do I tell my 12-year-old Italian student on Monday when he comes back for his lesson and point-blank refuses to believe that the bald eagle is bald? Can't say I blame him.

Question

1. Why was the white-headed eagle named “bald”?
   Is it because
   • its white head resembles a smooth spherical ball?
   • bald used to mean “streaked with white”?
   • bald used to mean “a white patch, blaze”?
   • because in flight it looks bald?
   • OR because its feathers are dark brown and white, and bald was/is a back-formation or clipping of the 16th-century term piebald.

Solution 1:

The earliest attested spelling of the word bald seems to be balled, as you have noticed. According to the Oxford English Dictionary (OED), "Middle English balled" is "of uncertain origin".

However, the use of "bald" to describe animals that have white streaks/spots/markings is attested well before the specific term "bald eagle". So it seems quite plausible that the bald eagle was simply named this because it was bald, as in, marked with white on its head. However, it is probably also not a coincidence that this white head resembles a hairless (bald) human head.

The OED gives it as the fifth definition:

5. Streaked or marked with white.

with the following examples:

• 1568 in W. Greenwell Wills & Inventories Registry Durham (1860) II. ii. 297 A little belled meare and a fole.
• [1594 R. Barnfield Affectionate Shepheard i. xxviii. sig. Bijv I haue a pie-bald Curre to hunt the Hare.]
• 1690 London Gaz. No. 2575/4 A black Mare with 3 white Feet, and a bald Face.
• 1711 London Gaz. No. 4848/4 Strayed..a black bald Gelding.

The American Heritage Dictionary (AHD) gives it as the fourth sense:

Zoology Having white feathers or markings on the head, as in some birds or mammals

So I would say, just tell your student that it is a specialized/obsolete use of "bald" that most speakers are not familar with.

This sense of bald also seems to be the origin of the word piebald (which occurs in one of the OED examples above). The AHD entry for piebald says:

PIE² [as in "magpie") + Early Modern English bald, marked or streaked with white.

Thus, bald eagle and the word piebald seem to be indirectly related. However, neither the OED nor the AHD seems to be in favor of describing "bald eagle" as having been directly derived from "piebald" (an etymology that Michael Seifert mentioned in a comment); i.e., I can't find any evidence that people used to call this animal a name like "piebald eagle" and then that was later shortened to "bald eagle".

Two etymologies that have been suggested for this sense of bald

The origin of bald in this sense seems especially unclear. It might be derived from the noun ball, which apparently doesn't have an attested cognate/ancestor in Old English but which seems to have cognates in some other Germanic languages. The OED says "ball" is

Probably the reflex of an unattested Old English *beall (compare Old English bealluc bollock n.), cognate with Middle Dutch bal ball, sphere (Dutch bal ball, sphere, ball of the foot, heel of the hand), Middle Low German bal ball, ball of the foot, heel of the hand, Old High German bal ball for playing, small sphere, mouthful (Middle High German bal ball, globe, ball of the foot, German Ball), Old Icelandic bǫllr ball, sphere, hill, Norwegian ball ball, Old Swedish balder, baller ball (Swedish boll, (regional) ball, balle), Old Danish bold ball (Danish bold ball), and further with Old High German ballo, pallo (weak masculine) ball of the foot, heel of the hand, mouthful, ball for playing (Middle High German balle ball, sphere, ball of the foot, heel of the hand), and Old High German balla, palla (weak feminine) ball for playing, mouthful < the same Indo-European base as classical Latin follis inflated ball, bellows, and (with different ablaut grade) ancient Greek ϕαλλός penis, image of the (usually erect) penis, especially as a symbol of the generative power in nature (see phallos n.). Compare bale n.3 for forms in Romance languages probably borrowed ultimately < the same Germanic base.

However, as you note, several sources suggest bald in the particular meaning "having white spots/streaks/a white spot/a white streak" is instead related to, or in some way also associated with Welsh "bal":

OED:

Compare Welsh ceffyl bàl a horse with a white streak or mark on the face (French cheval belle-face), where bàl may be an adjective, or a noun construed as a genitive.

AHD:

Sense 4 ["Zoology Having white feathers or markings on the head, as in some birds or mammals"] perhaps partly of Celtic origin; akin to Welsh bal, having a white streak on the forehead (of horses), Irish ball, spot, mark, and English blaze, white mark on the face of an animal; see BLAZE².

The word blaze is traced back by the AHD to the PIE root

‌‌ bhel-¹ To shine, flash, burn; shining white and various bright colors.

Which is identical in form with, but treated as a distinct root from

‌‌ bhel-² To blow, swell; with derivatives referring to various round objects and to the notion of tumescent masculinity.

Solution 2:

OED unequivocally derives the first morpheme in “bald eagle” (or “bald-eagle”) from the ordinary sense of bald as “Having no hair on some part of the head where it would naturally grow; hairless.”

In the entry for the bird, the following two extracts lead the shortish list:

1692 R. Frame Descr. Pennsylv. 27 The Turky-Buzard and Bald-Eagle high, Wild Ducks, which in great Companyes do fly.

1694 Philos. Trans. 1693 (Royal Soc.) 17 989 The Second is the Bald Eagle, for the Body and part of the Neck being of a dark brown, the upper part of the Neck and Head is covered with a white sort of Down, whereby it looks very bald, whence it is so named.

Count this slander upon Migizi among the many misnomers that we gichi-mookomaanag visited upon the creatures of Turtle Island, starting of course with Indians for Anishinaabeg and also including wild rice for manoomin.

And yes, “R. Frame” should definitely keep his day job—versification clearly is not his forte.

Solution 3:

Even 18th-century authors could not agree on why the bald eagle is so called.

From The Natural History of North-Carolina, printed in 1737

The Bald Eagle is the largest, and is so called, because his Head to the middle of the Neck is covered with a white sort of downy Feathers, whereby it looks very bald, and the Tail is as white as Snow, the rest of the Body being of a dark brown colour.

But in 1774 the British authors of The Beauties of Nature and Art Displayed in a Tour Through the World, claimed

In Virginia, Mr. Clayton informs us, there are three sorts of eagles; the largest of which they call the grey eagle, being much of the colour of our kite or glead: the second is the bald eagle, so called because the body and part of the neck and head are only covered with a whitish sort of down; and the third is the black eagle, much resembling those of our own island