음독 = literally "sound reading" = each hanja has one reading, each hanja's reading is a syllable in length, and it follows ancient Chinese [specifically Tang dynasty] rhyming laws (which were essential in the Confucian exam system for civil servants) and the grammar of Classical Chinese.
Therefore 弘益人間 being pronounced 홍익인간 is resolutely 음독.
훈독 = literally "meaning reading" = each hanja is mapped to a specific word which is (in some sense) more understandable. Their use is really to define each hanja, and so they are generally used with 음독; the most common use of this 음훈 system is with hanja names.
In this case, this would be spoken:
弘 클 홍, 益 더할 익, 人 사람 인, 間 사이 간.
Note that this is usually enunciated 음 - 훈 - 음, as if the 훈 was describing the second 음. The order of the original hanja is preserved, and no attempt is made to conform to standard Korean grammar - it is classical Chinese.
When this motto is then explained, the sentence is transformed into actual Korean to bring out the 뜻. The verb is moved to the end, relative clauses are formed, hanja characters are grouped. But as this involves translation into intelligible Korean, there isn't necessarily a pre-defined version; unlike the 음훈 which certainly does have a standard mapping.
In this case, the two explanations are both acceptable, and the differences come from how to interpret Classical Chinese into modern Korean.
- For Classical Chinese 人間 - do we have to bring out the meaning of the 間 with 세상 or not?
- Does a fully conjugated verb need to be inserted at the end? Or can we keep the tenselessness of the original Chinese?