This is my understanding of the development of the term jayanti -

Stage I – Jayantī = Śrījayantī

Presumably the older sense of Jayantī  is interchangeable with Śrījayantī . This latter term refers only to the utsava celebrating the birthday of Kṛṣṇa – the dark aṣtamī [or rohiṇi asterism] of the Śrāvaṇa/Bhādrapada month. This use of Śrījayantī  in this sense still prevails among the brahmana-s and other Hindu groups of Tamiḻ-speaking territories. This suggests that Śrījayantī was probably the archetypical jayantī ].

Besides the usage by Madhvacārya-Pūrṇaprajña (1238–1317), the term is attested in the ouvre of Veṅkaṭanātha-Vedāntadeśika 1268–1369. The first verse of Gopālaviṁśati repeated at the beginning of and Yādavābhyudaya uses the term jayantī unmistakably in the sense of  Śrījayantī .

vande brndāvanacaram vallavījanavallabhaṃ |

jayantīsambhavaṃ dhāma vaijayantīvibhūṣaṇam ||

The term is attested in a Siṁhācalam inscription of  Narahari Tīrtha (1243-1333) where it is not clear if the expression Śrījayantī  of śrīnarasiṁhanātha refers to the Kṛṣṇajayantī  or to the Narasiṁhajayantī .  

Stage II – The Birthday boom

The appearance of the term  jayantī  in an extended sense increases in the  inscriptions of  Vijayanagara times [~ 14^th century onwards]  and it is attached  to other deities - Kūrma, Narasimha, Rāma and Vāmana, for example.  

Stage III – Ossified in Government Parlance

Currently, the term is lexicalized in the lingua franca of Indian bureaucratese and applied to the remembrance days of all important (dead) personages -  Buddha, Mahāvīra, Śaṅkara-, Basava, Gandhi and Ambedkar  (illustrative list). There is also a Gītājayantī  to celebrate the ‘birth anniversary’ of the Bhagavad gītā.

The term in modern usage has the additional denotation of jubilee or anniversary. For example Svarṇa jayantī and Rajata jayantī for Golden Jubilee and Silver Jubilee.

Best,

Naresh Keerthi

Hebrew University, Jerusalem