You seem to think the studies are vague or just imaginary. They are not.
For example, 人 is the drawing of a man, 山 is the picture of a mountain. The relation is clear and you don't need to be very imaginative to draw the conclusions.
Same with ideograms in all ideographic languages, the relation of a symbol and its meaning is sometimes dead obvious. Do you call all these interpretations "vague" or "imaginary"?
Where do such interpretations come from?
Traces of etymology on single characters are seen on earlier classical texts, but the major source is the dictionary 说文解字, which is written in East Han dynasty, 1300 years after the first oracle bones. Later studies are largely based on this dictionary.
Are they based on older documented sources or are they later interpretations based on individual attempts of imagination and decomposition of a certain character?
Early documentation on the etymology of individual characters is scarce. There are individual attempts, probably a lot. The inventor of Chinese (if there is one) didn't leave any documents, so it's probably the only way to go.
Fortunately the Chinese writing system is largely ideographic, so the interpretations can't go very wrong. There is a set of general rules called 六书, it is a quite accurate description of the basic principles of creating Chinese characters.
The author of 说文解字 haven't seen a single oracle bone, but many of his interpretation can be verified by the later discovered oracle bones.
Are they authoritative in the sense that these explanations are widely accepted or they vary among different sources?
Later scholars have found errors in 说文解字, but it's still largely a credible source. The study of Chinese etymology is a mature subject. Competing theories on single characters do exist but rarely do you see it.
