Blood vessels

Arteries are defined by the direction of blood flow: they carry blood away from the heart to the different organs.
They cannot be defined by the type of blood they carry, because the pulmonary artery carries deoxygenated blood from the heart to the lungs.
A vessel that carries blood from one visceral organ to another is a portal vein (e.g., the hepatic portal vein).
Breaking up into capillaries that reunite to form a vein is not a defining feature; it does not distinguish an artery from a portal vessel.
Hence arteries are best defined as vessels that carry blood away from the heart to different organs.

A substance injected into a vein enters the blood stream directly; it is carried to the heart and then pumped to the whole body within a very short time. So intravenous injection gives the fastest distribution.
Veins are superficial, thin-walled and carry blood at low pressure, so they can be punctured safely and the bleeding stops easily.
Arteries are deep-seated and carry blood at high pressure; puncturing them is risky, and the material would reach mainly the organ supplied by that artery.
From muscles the absorption is slow (intramuscular injections are used for a slower or local effect), and lymph flows very slowly.
Hence the material should be injected into the veins.