May I ask what the evidence is that only the first three types were the the first classifications?
The evidence is mainly from comparative studies between different versions of Nikaya / Agama literature, particularly their structure.
cf. Yinshun 1971: 867ff; Nakamura 1980: 27-29 has some clear statements in English on this, without the analysis.
The first three angas are possibly (summarized from Orsborn 2010: 36ff):
i. Sutta / sūtra – most of the suttas in the nidāna-, dhātu-, ṣaḷāyanata-, vedanā-, khandha-, magga-, bojjhaṅga-, satipaṭṭhāna-, indriya-, sammappadhāna-, bala-, iddhipāda-, anāpānāsati-, jhāna- and sacca-saṃyuttas of SN (& SĀ) (cf. Yinshun 1980: 526f).
ii. Geya / geyya – verses, also known as gāthā and / or udāna, = Sagāthā-vagga of SN (& SĀ).
iii. Vyākaraṇa / veyyākaraṇa, two types: Tathāgata- and sāvaka-veyyākaraṇa. Equal the lakkhaṇa-, nāga-, supaṇṇa-, garuda-, valāhaka-, sotāpatti-, diṭṭhi- (and moggallāna-)saṃyuttas of SN (& SĀ) (cf. Yinshun 1980: 526f). Mainly in the khandha-vagga.
The best comparative work has probably been done by Yinshun (after Lü Cheng), and even Mizuno praises him for his efforts in this area. But you'll probably have to learn Chinese and maybe Japanese too, to read about this, because that will probably be faster than waiting for somebody to translate their work into English. Sorry about that. Maybe try Nakamura (1980: 28-29) as above.