I can't answer you by its history, for a large part is lost, and I fear that many of the scholastic articles are false information.
I tend to take a "conspiracy theory" perspective. I think there is some underlying motive to move the world to one direction, and religion/spirituality is one of their targets - to create a world religion that fits their goal. The rise of popularity or sudden trending of certain religious sect is a sign of caution. The Theosophical Society had and still has influence in Southeast Asia, particularly countries which once fell under colonial rule. A respectable Thai scholar turned monk advocating Thai Buddhist reform was under Theosophic influence; so was the Japanese Buddhist scholar D. T. Suzuki (his wife was a Theosophist). Sri Lanka was a lucrative English colony: a famous monk is also a politician, elite locals who were able to receive higher education or even migrated to Western countries, and thus had good English, were often the lineages the left-behind byproducts groomed by the colonial rulers to establish their rules. In China the popular monk Jing-kong, who teaches Namo-Amitabha as the one and only fix-all method to be reborn in Sukhavati: some evidence indicated he was once a spy, and was used by NSA to promote the 2012-GIG. Many are learning the interpretation of Sutta/Sutra from them, instead of learning directly from the Sutta/Sutra: this is worrying. Anyone with independent mind reads history, searches for the truth buried under history (or Hi[s]-Story) knew what Theosophical Society is, and how it participated in the World Wars. Those popular monks should be read with caution, are they searching for enlightenment, or, carrying a hidden agenda? (Note 1 and more notes if needed)
Now laying clear of above, what are these schools? You are right, Buddha didn't create schools. Since no one is as completely enlightened as the Buddha those students had different understandings of his teachings, all are incomplete, especially one schooled by the schools. To summarize these schools, I would instead differentiate on what their paths lead to:
A) Hinayana: for self-liberation, the highest goal is Nirvana/Nibbana, Arahat.
B) Mahayana: for Buddha-hood. To realize Buddha-hood a practitioner adopts the Bodhisattva Path to cultivate wisdom and merits, to reach perfection of Prajna and untainted conducts. This could take for many Kalpas. After realizing self-liberation a Bodhisattva arose from Nirvana reborn in the Samsara to further wisdom quest and helping others to realize The Path. The apex is attaining Anuttara-samyak-sambodhi.
C) Vajrayana: nowadays represented by Tibetan Buddhism, for immediately realizing Buddha-hood by mastering the subtleties of the human body or, energy (an not exact term). These [must at the final stage] involve Tantric practice (the learnt should know what is Tantra Yoga), and the mouth-to-ear transmission of secret teachings from Guru to disciple. I have to declare my understanding maybe partial. There may be some other teachings but what I stated is valid too.
Theravada vs Hinayana, The Pali Suttas
Although the modern Theravadist
footnote 1 tries to decline the relation to Hinayana it seems their final goals are the same: i.e. Self-liberation & Nibbana. Because the Theravadist has only the Pali Canon, in which the Suttas are the four Nikayas, and the rest are Abidhammas (treatises written by Bhikkhus not directly Buddha's teachings) their canon don't include teachings about how to reach Buddha-hood, nor what is Buddha-hood. Instead of asking whether there may be missing teachings in their Suttas, some conventional Theravadists disparage the Sutras of other schools, saying that they not from the Buddha but composed by some later monks. I think it's worth investigating how the Pali Canon originated: obviously Buddha didn't visit the island of Sri Lanka. I read in this forum Soumen quoted a claim that it was bought there by some monks sent by Mahendra: these monks took what they only could have at that time. However, even the Mahinda/Mahendra legend is suspected a pure Sri Lankan invention due to the obvious fact of lacking any historical artifacts left behind of such supposed important event.
The latest discovered fragments of a Mahayana Sutra dated 75CE are almost identical with the Chinese version. In contrast, the Chinese Classical Sutras have as background that Bhikkhus went to India (i.e. "The West") to obtain them, or that Bhikkhus of India went to China (The Middle Kingdom or Middle Earth, 中土) taking with Sutras with them. The Chinese Sutras kept complete teachings related to almost all the schools; the Vinayas are a collection of different schools kept separately in their original formats, not combined to produce any defined monastery rules. Many of the modern Sanskrit Sutras are back-translated from Chinese, since these Sutras were lost in India. However conventional Theravadists, allied with some modern Western scholars (Edward Conze was an old Theosophist), and some Western Theras (monks), claim (unjustly, in my opinion) that the Pali Canon is the "only" Buddhavacana, the oldest and purest, and discredit the rest of the Buddha's teachings recorded in other Sutras. Some Theravadists adopt Western scholars' interpretations of the Suttas, instead of understanding in original Pali by tradition: a lot of the English translations of Pali Canon are of doubtful accuracy.
In this sense the conventional Theravadist is indeed not the same as Hinayana. "Hinayana" means someone taking an immediate mean to liberate himself, reaching Arahat-hood. Hinayana doesn't label Sutras other than Agamas fake, but just that they don't see the Bodhisattva Path is feasible. Agama Sutras, which are equivalent to and cover the entirety of the Pali's Nikayas (Suttas), are among the many to be studied. Agama Sutras recorded only fragments of the teachings and very repetitive, instead those most centred in meditation techniques and how to transform the Vijnanas to free from the grip of the Four Greats - the body are studied
footnote 3. The Sutras for the path of Hinayana are also collected in the Chinese Tripitaka; whilst the traditional (old) Theravadist may have to rely heavily on the Abidhammas such as the Visuddhimagga written by Buddhaghosa
footnote 4 to practice meditation for self-liberation.
I expect that some may disagree with my answer, but I thank you for letting me voice it here.
^The reason to give emphasis on modern is, according to historical records and scholastic studies some sub-divisions of original Theravada practiced Mahayana Doctrines.
^In the accounts of Ashoka, Mahendra his son a Bhikkhu never mentioned, it was missing in all the Ashoka edicts.
^For Hinayana practitioners: 《正法念處經》(highly recommended for those have the taste of the Nikaya/Agama style),《成实论》,《俱舍论》... etc.
^Buddhaghosa originally was a Brahmin scholar, his commentries shown he was influenced by Mahayana and Yogacara. He might have visited China at his time when China was the capital of Buddhism, but due to his old age he didn't set foot Canton instead returned on the same boat.