The OLD says: liberi "sons and daughters, children (in connection with their parents)."
First of all, it is important to remember that, as Osgood 2011 puts it,
"... it was of great concern to determine the legal status of any children born: Romans guarded citizenship rights jealously" (p. 76; emphasis mine - Alex B.).
I'm no expert in Roman law but it seems to me liberi quite often implied legal descent and hence inheritance rights, cf.
"At parentes, si pergunt liberi errare, bonis exheredant" (Gellius)
or the very phrase liberi naturales (legitimized children in concubinatus, who could become legitimi in the absence of other legitimate children and their mother must have been free-born),
or the famous law, Ius trium liberorum.
In Senatus consultum de Asclepiade Clazomenio sociisque (78 BC) we read the following:
“... whatever inheritances have come to them or their children by chance, / these are to fully hold, possess and enjoy; whatever lawsuits they, their children, their descendants or their wives may bring against another person, or / if other persons bring lawsuits against them, their children, their descendants or their wives, these men, their children, or their wives are to have the right of choice …” [English translation by Andrea Raggi; Raggi 2001]
In pre-classical Latin inscriptions traces of the original diphthong can be seen, e.g. LEIBEREIS (oi > ei, cf. Weiss 2009/2011, pp. 102-103).
Wiedemann 1989/2014 summarizes it nicely:
"The most common word, liberi, is associated with the concept of libertas, 'freedom', not in our western liberal sense of being independent of others, but in the sense of being a member of the (free) community; it is philologically cognate with the Greek word for freedom, eleutheria, but also with the German word for 'people', Leute. The liberi were on the one hand those junior members of a household who were free, as opposed to the slaves, servi; and on the other hand a collective group of free-born Roman boys and girls, contrasted with adults of citizen status. They were the future citizen community.
The other common word for child puer, refers to the junior members of the family or household; in the classical period, the same word is used for the free-born children of the paterfamilias, and for his slaves, whatever their age."
TLL (Thesaurus linguae Latinae) confirms this: