First you make the statement that - "However, there are Hindus who believe that there really is the first birth of a jiva in this material world, and thus they do not believe in infinite number of rebirths since infinite past, and thus they do not believe in beginningless samsara." - I am not aware of any scripture that supports this statement of yours. Any Hindu that has this 'belief' is not orthodox as it is not supported by scripture.
Krishna says in the Gita (II.12) Swami Nikhilananda translator:
Never was there a time when I did not exist, nor you, nor these kings of men. Never will there be a time hereafter when any of us shall cease to be.
And in Brahma Sutras 2.1.35 (Swami Vireswarananda translator):
If it be said (that is) not (possible) for want of any distinction in work (before creation), (we say) no, because of (the world) being without a beginning.
and Sankara's commentary on this verse:
Since before the first creation the individual soul [jiva] cannot possibly have had a previous existence, whence comes the difference in the condition of beings in that first creation, unless the Lord has caused it out of His partiality? the objection is answered by the Sutra, which says that creation is without a beginning and the question of first creation cannot arise. It is like a seed and its sprout. So the individual soul, should have always had a previous existence and done good or bad deeds in accordance with which their lot in a subsequent creation is ordained by the Lord.
And Ramauja's commentary on this verse from his Sri-Bhasya (Swami Vireswarananda translator):
Before creation there was Brahman alone and noting else existed; there were no individual souls and so there was no Karma to justify the inequality in creation. That there were no souls before creation is declared by the scriptures: 'In the beginning, dear boy, there was this Being alone, one only' etc. (Cha. VI. ii. 1). The Sutra refutes this and says 'No'; for the souls and their Karma form an eternal stream which is beginningless. Individual souls are not created but existed even before creation in a very subtle condition almost non-distinguishable from Brahman, and hence the scriptural texts which declare the non-existence of everything but Brahman before creation. What the texts deny is the existence of beings in a gross state with name and form. But the souls did exist in a subtle condition before creation. This is reasonable also for otherwise souls would be punished for acts they did not commit and go without punishment for wrong acts committed by them. That the souls are eternal is also declared by the scriptures: 'All this was then unmanifest. It became manifest only as name and form' (Br. I. iv. 7). As the text talks of mere manifestation in gross form, the souls are eternal and existed before creation also. 'He is the eternal among the eternal' (Sve. VI. 13); 'This unborn, eternal, everlasting ancient One' etc. (Ka. I. ii. 18); 'The conscious sybject and the unconscious object, the master and the dependent are both unborn' (Sve. I. 9). Smriti also declares the same: 'Know that Prakrti and Purusa are both beginningless' (Gita XIII. 19). Moreover, creation also is beginningless, and when the scriptures talk of the beginning of creation they mean only the beginning of a new cycle. This is borne out by texts like, 'The Lord devised the sun and the moon as before' (Rig Veda X. cxc. 3).
So partiality and cruelty cannot be attributed to the Lord.
There are an infinite number of jivas and there is an infinite past and an infinite future. Infinite means infinite, without number.