The Latin suffix -io, stem -ion-, is usually added to the supine stem ( ~= past participle stem) of a Latin verb, in order to form a noun of action (meaning "x-ing"). It is sometimes also added to other stems, but usually not. The verb vocare ("to call"), present stem voca-, supine stem voca-t-, gets vocatio ("calling"); the stem of that noun is vocation-, which is what our suffix was based on.
Because the regular suffix for the supine stem / past participle is -t- in Latin, we often see -tion; but compare also miss-ion, fus-ion, cohes-ion, inflex-ion, etc.. The same suffix can also be found with non-supine stems, though less frequently so, like rebellion, legion, etc.
Now how can we predict what comes before -ion in English? I don't have a neat system based on tight rules, but perhaps I can give a few general directions.
What we could do is try and think of a cognate verb, as you have been rightly doing. Does this verb end in vowel + t or vowel + te? If yes, it is probably based on a (real or reconstructed) Latin supine stem; then -ion can come right after the t, as in inhibit, complete, migrate.
And what if the verb does not end in vowel + t(e)? Then you need to know whether or not its stem is a supine stem in Latin. There is no way to know this except by consulting a Latin dictionary. But there are some unreliable rules.
Irregular supine stems most often end on s or x, so verbs like flex (from verbal stem flect-, "to bend") and fuse (from verbal stem fund-, "to pour") contain the supine stems flex- and fus-, leading to inflexion and fusion. But vex- and pos- are not supine stems, so vexation and position, from supine stems vexa-t- and pos-i-t- (the i is probably a fused theme vowel).
There are many supine stems that end on -pt- or -ct-, like act- and rupt-, leading to action and eruption.
Verbs ending on -nt are nearly always based on the present-participle stem -nt-, and so cannot get -ion right after nt; but there are a few supine stems on -nt- as well, mainly vent- and tent-, leading to -vention and -tention.
But there are also some nominal ("noun") stems that can get -ion, as mentioned before, like mens, "mind", stem ment-, leading to mention; and dens "tooth", stem dent-.
Cohabitation is an apparent exception. But both habē-re ("to have") and habita-re ("to inhabit") exist as verbs. Habitare was actually formed based on the supine stem habit- from habēre; then a new supine stem was formed based on this new verb, habitat-. And so we have pro-hibi-t-ion and co-habi-t-at-ion.
Normally it isn't very functional in Latin to make a new verb based on the supine stem of another, since you already have the original verb; but it is sometimes done, often to add a sense of frequency or intensity, and so two supine forms may come to exist. Usually a is then added after the supine stem to turn it into a new verbal stem, because a indicates a causative verb, i.e. a verb that means not "to do x" but "to cause someone to do x" (with nominal stems, "to turn into x, to affect with x", or simply to turn any nominal stem into a verb). So fugere = "to flee"; fugare = "to make flee, to drive away"; donum = "gift", donare = "to present as a gift" (or simply "to give").
Cf. haerē-re ("to stick") => hae-s-us (past participle, "stuck") => hae-s-ere => hae-s-it-us => hae-s-it-are => hae-s-it-at-us => hae-s-it-at-io =>> hesitation. Notice that the same supine suffix has been added thrice: s, it, at (the t was turned into an s after certain verbal stems; the i and the a are theme vowel and causative vowel, respectively). And so we have co-he-s-ion and he-s-it-at-ion, both from the same verbal stem haer(e)-.
So it is theoretically possible to add -ation after any supine stem; but that is usually not done, because just -ion is shorter. After any stem that isn't a supine stem, just adding -ion is unusual, so we will usually have to turn it into a (real or hypothetical) verb first with -a-, then add the -t-, then -ion — i.e. we have to add -ation. But, as mentioned above, there are many exceptions, where -ion can be added directly to non-supine stems; however, this is a fixed set of words that mostly already existed in that approximate form in Latin.
Best Answer
I haven't found any common factor distinguishing the elements with names ending in -ium from the elements with names ending in another letter followed by -um. It seems fairly arbitrary, and in fact there was some variation historically between forms ending in -ium and -um for tantalum and lanthanum (and there is still variation between aluminium and aluminum in American English).
Most elements are metals. Many of the nonmetals are halogens or noble gases, which have different naming patterns using the suffixes -ine and -on respectively. But the nonmetals helium and selenium, and the metalloids tellurium and germanium have the -ium suffix also. The user J... left a comment saying that helium was originally thought to be a metal; a quote supporting this can be found in Linear Christmas's answer to the related Chemistry SE question Why do the names of most chemical elements end with -um or -ium?. The Oxford English Dictionary (OED) entry for selenium says that it and tellurium were "formerly classed among the metals".
Apparently, in Latin the names of metals always had the neuter grammatical gender ( Lectures on Syntax: With Special Reference to Greek, Latin, and Germanic, by Jacob Wackernagel, edited by David Langslow, published 2009). Examples of this are ferrum "iron," aurum "gold," cassiterum "tin" (a loanword from Greek κασσίτερος). Latin neuter nouns of the second declension regularly take the ending -um when singular and in the nominative or accusative case. (Janus Bahs Jacquet has pointed out in a comment that some metal names in Latin belonged to other declensions and so took other endings, such as aes "copper/bronze/brass." But I have the impression that new words formed in scientific Latin are generally in the first or second declension.)
So the -um part of -ium is just this Latin second-declension neuter ending. In Latin, -i- is a common derivational suffix (words formed with the suffix -i- end in -ia/-ius/-ium depending on their gender). Generally, Latin -i- is used to derive adjectives from other nouns (Wiktionary) but in Latin adjectives and nouns can often interchange, so it's often also possible to use words with this suffix as nouns. I haven't found a reference that explains why -i- shows up in so many of these element names; I'd assume it's because as new elements were discovered people named them in honor of existing things and the easiest way to derive a new name like this in Latin is to use -i-. For example, rhodium is apparently named after roses (the Ancient Greek word for "rose" is rhodon). Also, as more and more elements were named this way it became a noticeable pattern that people might follow even if they were not aware of its origins.
The OED doesn't provide an extensive explanation of the origins of the use of -ium, but it mentions the Latin metal names and gives some historical context for the English ones:
As I said, I can't see any factor connecting all of molybdenum, platinum, tantalum, and lanthanum. (The names tantalum and lanthanum were coined more recently than molydenum and lanthanum, so I have re-arranged your list to put them last.)
Molybdenum, according to Wiktionary, is derived from the Greek name for lead, μόλυβδος molybdos. The OED provides some additional details about how the -en- got in there: it references the earlier word "molybdena", from Latin molybdaena, from ancient Greek μολύβδαινα molybdaina "a kind of lead ore" which according to the OED ends in "-αινα, suffix forming nouns". The rule about metal names having neuter gender in Latin does not seem to apply to the names of ores: there are also words like galena (feminine) "lead-ore; dross", anthrax (masculine) "cinnabar", and (lapis) haematites (masculine) "red iron-ore/hematite" (it also had other meanings, apparently). I don't see any particular reason for why molybdenum was used as the name of the element instead of molybdenium.
Platinum seems to be from a Spanish word, platina, meaning "little silver." I guess it makes some sense to adapt this by simply changing the ending to -um rather than deriving a new word with -i-. But the process seems fairly arbitrary.
Tantalum (discovered and named in 1802) is named after Tantalus. I don't see any reason why it isn't tantalium. In fact, Wiktionary lists that as an obsolete spelling, and there is a question that was asked earlier on this site about that variant of the word: Reason why tantalium became obsolete.
Lanthanum (named sometime around 1840) is named after a Greek word meaning "to go unnoticed." I don't know why it doesn't have the ending -ium; in this case as well, it seems completely arbitrary to me. Wiktionary lists lantanium and lanthanium as obsolete alternative forms, and there is a question about them here: Lanthanum vs lanthanium.