This is a very interesting question. Unfortunately, I have not been able to find enough information to give more than a partial answer. But I hope it will be of some use.
According to the Oxford English Dictionary (OED), the prefix "pro-," meaning "before," comes mainly from the Greek prefix προ-. It is cognate with the Latin preposition pro meaning "for, before, in front of," and with the English word fore. More than 60 words with this prefix were adopted into Latin in antiquity, including prologus "prologue" and propheta "prophet," so Latin has been an important intermediate source of words with this prefix.
The OED indicates that the Latin prefix prae-, which corresponds to modern English pre-, was mainly used in Classical times with verbs and verbal derivatives, or to intensify the meaning of adjectives. There are also some cases where it modifies the meaning of nouns and adjectives, but the use before a noun to indicate an earlier version of that noun apparently only became common in post-Classical Latin and later languages.
The prepositional construction, in which pre- governs the second
element, which was so rare in Latin, has in English received vast
extension (as also in French), pre- being preferred to ante- as the
opposite of post- in new formations, and often substituted for it, as
in pre-Christian, prehistoric, pre-Darwinian, pre-reformation
instead of ante-Christian, ante-historic, ante-Darwinian,
ante-reformation. [...] It may also be combined with nouns in order to form compound nouns directly, as precancer n., preclimax n., [...] premyelocyte n., prepuberty n., etc. Compounds of this type are of relatively recent appearance.
So the distinction you mention does not seem to date back to Classical languages.
It does remind me of the use of hyper- in some contexts as a more extreme version of super- (discussed in the answers to this question: Which is higher — "hyper-", "ultra-" or "super-"?). In both cases, it seems like the more common term (super- or pre-) is used for the less extreme meaning. I have not found any cases where pre- and pro- are distinguished this way outside of biology.
I found one document about biology which seems to describe a slightly different naming scheme using "pre-pre-" instead of "pro-":
So-called B-ALL, for example, is based on progenitor cells of
B-lymphocytes, while T-ALL forms from precursors of T-lymphocytes. A
degeneracy in the early development stages is characterised by the
prefix "pre". This results in the following ALL-subtypes:
- Pre-pre-B-ALL (now commonly referred to as pro-B-ALL)
- Common ALL
- (mature) B-ALL
- Pro- and Pre-T-ALL
- Intermediate (cortical) T-ALL
(Acute lymphoblastic leukaemia (ALL) - Brief Information)