The pattern seems to be based on word endings, not just on word roots
I think that the examples that you mention actually can be explained synchronically for the most part, if we look at the morphological structure of the words. So even though you asked about history, I will take a bit of a detour in the first part of this answer to talk about the behavior and distribution of the negative prefix in- in present-day English.
Whether the root of the word is from Latin or from English (Germanic) doesn't actually seem to be as relevant as whether the ending of the adjective is from Latin or from English.
The negative prefix in- is not very productive in English1: that is, it's rare for a speaker to take an English adjective and attach in- to it to create a new negative adjective. Rather, in- (or im-, ir-, or il-) mainly shows up on words that already existed with this prefix in Latin or French before they were adapted for use in English. For example, before the word instability was used in English, instabilité was used in French.
The adjectives moved, determined and varied were derived in English by the addition of the native English suffix -(e)d to the English verbs move, determine and vary. The suffix -(e)d is a sign that these words were not taken directly from Latin adjectives.
In contrast, adjectives ending in “-ble”, “-ate” and “-ant” often do come from Latin adjectives (the corresponding Latin endings in the nominative masculine singular are -bilis, atus and -ans), and consequently, many adjectives with these endings have corresponding negative adjectives starting with in-/im-/ir-/il-.
The reality is a bit more complicated than my description in the preceding paragraphs—see my answer to the older question "Why do we say INcomplete but UNcompleted?" for some more details and citations. But it seems to be fairly clear that if you look at a negative adjective ending in the suffix -ed, it is much more likely to be prefixed with un- than with in-, regardless of whether the adjective is built on a Germanic or Latin root.
Historically, un- is the older suffix
Of course, English is a Germanic language, and so the Germanic prefix un- has a longer history of use in English than the prefix in-, which is from Latin and French.
I don't know yet how the negation of adjectives with Latin or French roots has evolved over time.
The Oxford English Dictionary (OED) entry for unstable has a citation dated a1340 and one that is more tentatively dated to "?c1225". The earliest OED citation for innocent, with the Latin/French negative prefix, is from "a1340". So it seems safe to say that during the Middle English time period, words starting with negative in- entered the language, but the native English prefix un- was already being used on some words of Latin or French origin.
The OED entry for the negative prefix in- suggests that its usage has changed some over the centuries, and mentions the substantial overlap with the usage of un- which has resulted in some amount of variation for a number of words:
In English in- (il-, im-, ir-) is a living negative suffix for words of Latin or Romanic origin, freely used, even when no corresponding formation appears in Latin; in this use it interchanges to some extent with the Old English negative un-, which is used in native or thoroughly naturalized words, e.g. incautious, uncautious, in-, un-ceremonious, in-, un-certain, in-, un-communicative, in-, un-devout, in-, un-distinguishable. In such cases the practice in the 16th and 17th c. was to prefer the form with in-, e.g. inaidable, inarguable, inavailable, but the modern tendency is to restrict in- to words obviously answering to Latin types, and to prefer un- in other cases, as in unavailing, uncertain, undevout.
Keep in mind that this entry "has not yet been fully updated (first published 1899)" and it may have some inaccuracies. I think the author may have been overstating things a bit with the wording "freely used": Dixon, who I cited above and link to below, says in- is "seldom used to derive new [words]" (p.73) and categorizes it as "scarcely productive" (p. 73-74).
I will try to find a more modern academic source to cite about the history of the prefix in- in English.
Citation footnotes & links
- Making New Words: Morphological Derivation in English,
by R. M. W Dixon (2014), p. 73-74.