According to Sanskritdictionary.com, the word, one meaning of 'JAti' is:
जातिः f. [जन्-क्तिन्] : A caste, tribe or class (of men); अरे मूढ जात्या चेदवध्यो$हं एषा सा जातिः परित्यक्ता Ve.3; (the primary castes of the Hindus are only four :-- ब्राह्मण, क्षत्रिय, वैश्य and शूद्र.
And One meaning of 'VarNa' is :
वर्णः [वर्ण्-अच् Uṇ.3.1] : A class of men, tribe, caste (especially applied to the four principal castes, ब्राह्मण, क्षत्रिय, वैश्य and शूद्र; वर्णानामानुपूर्व्येण Vārt; न कश्चिद्वर्णानामपथमपकृष्टो$पि भजते Ś.5. 1; R.5.19.
So both have identical meanings as per the dictionary.
As has been pointed out elsewhere in other posts, Arjuna mentions in the Gita that the 'VarNa-sankaras' ruin the family because they are not entitled for the 'Pitri-karma'-s (1/40-43)
1.40 From the ruin of the family are totally destroyed the traditional rites and duties of the family. When rites and duties are destroyed, vice overpowers the entire family also.
1.41 O Krsna, when vice predominates, the women of the family become corrupt. O descendent of the Vrsnis, when women become corrupted, it results in the intermingling of castes.
1.42 And the intermingling in the family leads the ruiners of the family verily into hell. The forefathers of these fall down (into hell) because of being deprived of the offerings of rice-balls and water.
1.43 Due to these misdeeds of the ruiners of the family, which cause intermingling of castes, the traditional rites and duties of the castes and families become destroyed.
Sri Krishna also mentions that the 'Varna-sankara'-s ruin the world (3/24):
3.24 These worlds will be ruined if I do not perform action. And I shall become the agent of intermingling (of castes), and shall be destroying these beings.
Unless Varna is determined by birth, these opinions of Arjuna and Krishna lose meaning.
Sri Krishna also uses the term 'PApa-yoni'(Gita,9/32) which is again an indicator of His relating Varna to birth.
So it seems that according to the Sanatana Dharma, the two words 'Varna' and 'JAti' have identical meaning.
I will be happy to get corrected by authentic sources.
I have not emphasised on Jnati and Kula hitherto as their meanings are straightforward.
The words 'Jnati' means 'relatives' and 'Kula' means to 'family' as opposed to castes.
Some dictionary-meanings of 'Jnati' are:
ज्ञातिः [ज्ञा-क्तिच्] 1 A paternal relation, a father, brother &c.; agnate relatives collectively. -2 A kinsman or kindred in general. -3 A distant kinsman who is not entitled to the oblations offered to deceased ancestors. -4 A father. -Comp. -कर्मन् n., -कार्यम् the duty of a kinsman. -चेलम् A low-born person; विभिन्न- कर्माशयवाक् कुले नो मा ज्ञातिचेलं भुवि कस्यचिद् भूत् । Bk.12.78. -प्रायः A meal for kinsmen (Mar. जातिभोजन); प्रक्षाल्य हस्ता- वाचम्य ज्ञातिप्रायं प्रकल्पयेत् Ms.3.264. -भावः kin, relationship. -भेदः dissension among relatives. -विद् a. one who has or makes near relatives. etc
Some dictionary-meanings of 'Kula' are:
कुलम् 1 A race, family; निदानमिक्ष्वाकुकुलस्य संततेः R.3.1. -2 The residence of a family, a seat, house, an abode; ददर्श धीमान्स कपिः कुलानि Rām.5.5.1; वसन्नृषि- कुलेषु सः R.12.25. -3 A high or noble family, noble descent; कुले जन्म Pt.5.2; कुलशीलसमन्वितः Ms.7.54,62; so कुलजा, कुलकन्यका &c. -4 A herd, troop, flock; collection, multitude; मृगकुलं रोमन्थमभ्यस्यतु Ś.2.6; U.2.9; अलिकुलसंकुल Gīt.1; Śi.9.71; so गो˚, कृमि˚, महिषी˚ &c. -5 A lot, gang. band (in a bad sense). -6 A country. -7 The body. -8 The front or fore part. -9 A tribe, caste, community. -1 A blue stone. -लः The head of a guild or corporation. etc.