Why do we "shed" blood, sweat or tears but not other things?
I found the following definition of shed (the verb):
chiefly dialect : to set apart : segregate
to cause to be dispersed without penetrating
a. to cause (blood) to flow by cutting or wounding
b. to pour forth in drops shed tears
c. to give off or out sheds some light on the subject
to give off, discharge, or expel from the body of a plant or animal: as
a. to eject, slough off, or lose as part of the normal processes of life a caterpillar shedding its skin, a cat shedding hair, a deciduous tree sheds its leaves in the fall
b. to discharge usually gradually especially as part of a pathological process shed a virus in the feces
to rid oneself of temporarily or permanently as superfluous or unwanted shed her inhibitions, the company shed 100 jobs
Here, I am primarily interested in the third usage. We can shed blood, sweat and tears but not much else. Obviously, we can also shed our clothes or shed light upon but these are different meanings.
The etymology of the word is:
shed (v.) "cast off," Old English sceadan, scadan "to divide, separate, part company; discriminate, decide; scatter abroad, cast about," strong verb (past tense scead, past participle sceadan), from Proto-Germanic *skaithan (cf. Old Saxon skethan, Old Frisian sketha, Middle Dutch sceiden, Dutch scheiden, Old High German sceidan, German scheiden "part, separate, distinguish," Gothic skaidan "separate"), from *skaith "divide, split."
According to Klein's sources, this probably is related to PIE root *skei- "to cut, separate, divide, part, split" (cf. Sanskrit chid-, Greek skhizein, Latin scindere "to split;" Lithuanian skedzu "I make thin, separate, divide;" Old Irish scian "knife;" Welsh chwydu "to break open"). Related: Shedding. A shedding-tooth (1799) was a milk-tooth or baby-tooth.
In reference to animals, "to lose hair, feathers, etc." recorded from c.1500; of trees losing leaves from 1590s; of clothes, 1858. This verb was used in Old English to gloss Late Latin words in the sense "to discriminate, to decide" that literally mean "to divide, separate" (cf. discern). Hence also scead (n.) "separation, distinction; discretion, understanding, reason;" sceadwisnes "discrimination, discretion."
As far as I can tell, the third meaning of shed (in the quoted definition) is restricted to blood, sweat and tears. Why is that? What is the origin of the idiom to shed blood? I would guess that shed blood comes from the meaning to scatter abroad of the Old English word sceadan, if so, why is it so restricted today? Was it once a more common term? Could we once say that I shed water on my garden or I shed the seeds in my field?
Because these usages of shed are assuming a few things about the objects being shed:
a. to cause (blood) to flow by cutting or wounding
b. to pour forth in drops shed tears
Both (a) and (b) require a liquid state in order to flow or drop and (c) is some form of luminance which you noted you don't actually care about. So you could say:
(a) The cyborg shed oil from its veins.
(b) The sky shed rain upon the fields.
These are non-standard in the sense that their usage is extremely uncommon but the meaning still fits.
To directly answer your question: You can shed blood, sweat and tears because they are liquids dispersed from cutting or wounding (blood) or things that pour forth in drops (sweat; tears). If anything else in the human body could do either of those things you could also shed them.
To prove the point:
A urinary tract infection has been plaguing me for days. Yesterday I shed a mere three drops.
By the way, I have no idea where you copied your definitions from but the link you gave doesn't seem to match.