Can anyone shed any light on the splitting?
In the same way as Martin I haven't been able to find a reference to "seven souls" within Buddhism.
In vaguely related traditions,
Hun and po
Hun (Chinese: 魂; pinyin: hún; Wade–Giles: hun; literally: "cloud-soul") and po (Chinese: 魄; pinyin: pò; Wade–Giles: p'o; literally: "white-soul") are types of souls in Chinese philosophy and traditional religion. Within this ancient soul dualism tradition, every living human has both a hun spiritual, ethereal, yang soul which leaves the body after death, and also a po corporeal, substantive, yin soul which remains with the corpse of the deceased. Some controversy exists over the number of souls in a person; for instance, one of the traditions within Daoism proposes a soul structure of sanhunqipo 三魂七魄; that is, "three hun and seven po". The historian Yü Ying-shih describes hun and po as "two pivotal concepts that have been, and remain today, the key to understanding Chinese views of the human soul and the afterlife."
The above seems to be seven bodily souls and three spiritual souls, which is more-or-less what you said in the OP.
The same articles says,
Hun 魂 and po 魄 spiritual concepts were important in several Daoist traditions.
... and I think there's some historical connection between Daoism and Buddhism in China. Signification of Buddhism and Taoist influences says,
When Buddhism came to China, it was adapted to the Chinese culture and understanding. Theories about the influence of other schools in the evolution of Chan are widely variable and rely heavily on speculative correlation rather than on written records or histories. Some scholars have argued that Chan developed from the interaction between Mahāyāna Buddhism and Taoism, while others insist that Chan has roots in yogic practices, specifically kammaṭṭhāna, the consideration of objects, and kasiṇa, total fixation of the mind. A number of other conflicting theories exist.
So maybe if you do "clearly remember reading somewhere about some branch of Buddhism", then possibly it was a branch with a Chinese influence.
What is seven constituent parts of the human soul?
From Theosophy by Rudolf Steiner
- Physical body
- Life Body
- Astral Body
- The "I" as the soul's central core
- Spirit Self as transformed astral body
- Life Spirit as transformed life body
- Spirit Body as transformed physical body
If the above is an accurate quote, maybe (who knows) it might be a match for your OP.
And if I understand correctly the hints in this answer (I don't know much more about it than that), the Theosophists wanted to define or synthesize a "universal religion", perhaps by including bits from Buddhism.
I think there have been other historical connection between Buddhists and Theosophists, so perhaps you were reading something Theosophical and conflating it with Buddhism somehow.
And is my current understanding correct, incorrect or oversimplified?
I think that Buddhism traditionally teaches that there isn't any (permanent) 'soul' at all. The/a non-Buddhist doctrine was/is of Atman (Self) and Brahman (Divinity), and (in contrast to that) the Buddhist doctrine was/is, instead, of Anatman.
I think that an example of that is as stated in this recent answer of mine about Milinda, Nagasena, and the parable of the chariot.
I find it a bit metaphysical: talk of karma and rebirth and so on.
There are a lot of topics about that kind of thing on this site, for example, anatta, and rebirth.
Instead some people prefer to focus on the practice of Buddhism within 'this life'.