Arguing about correctness in this area seems fairly pointless to me since it's all a matter of preference and opinion. The word element metr-/metre/meter has the same ultimate source in all of these words: Greek μέτρον (corresponding to Latin metrum and French metre).
Since the time of John Walker at least (1791), people interested in pronunciation have been aware of a commonly observed rule about stress in compound words derived from Greek sources. It goes something like this: accent the antepenultimate (third-to-last) syllable, as in rhinology or orthography, unless the penultimate (second-to-last) vowel is followed by a consonant cluster, as in rhinoplasty or orthodoxy (there are also other exceptions that I don't know how to summarize).
This rule is also mentioned by the linguist John Wells in one of his blog entries:
Every adult native speaker knows the words paralysis and analysis. Some also know one or more of catalysis, dialysis, electrolysis or some fifty other technical words with the same ending. All have antepenultimate stress: /pəˈræləsɪs, əˈnæləsɪs, kəˈtæləsɪs, daɪˈæləsɪs, ˌelekˈtrɒləsɪs/ etc.
Two years ago I was diagnosed with a heart condition and underwent an angioplasty procedure. Since then I have attended the Cardiac Support Group run by my local hospital. One of the techniques for treating a heart attack is known as thrombolysis. But I have noticed that the cardiologists, cardiac surgeons, paramedics and nurses don’t call this /θrɒmˈbɒləsɪs/, as you would predict from all the other -lysis words. They call it /ˌθrɒmbəʊˈlaɪsɪs/. Ah well, language doesn’t obey rules as much as some linguists would like.
In my lectures I sometimes talk about allophony. Guess how I pronounce it. Remember that everyone agrees that telephony is /təˈlefəni/ and cacophony is /kəˈkɒfəni/, and that I am someone who plays by the rules.
The origin of this rule is not clear to me, but there are many examples, such as microscopy and all of the -ologies, -onomies and -ographies.
This rule is the origin of the stress pattern of pentameter, tetrameter etc, as well as thermometer, seismometer.
The stress pattern used for metric terms reflects a countervailing tendency towards pronouncing Greek compounds using the rules applied to normal English compounds: no stress shift, secondary stress on the second element in the same place where it would be in the independent word. This tendency is also old (you can see at the link that Walker mentions it, to criticize it), but it's possible that it gained more influence on the pronunciation of metric terms than on other words because it seems to many people a more "logical" way of pronouncing things, and the metric system is all about doing things more logically than traditional systems.
Also, rules that make variation meaningful by explaining it with some kind of semantic distinction (like the one you give in your post, "words stressed the latter way usually refer to equipment; speedometer, odometer, thermometer, seismometer, etc. Words stressed the former way are used for the metric measurements: millimetre, centimetre, kilolitre, etc") tend to be popular for some reason. So once a distinction arises, even if there's no particularly good reason for its origin, prescriptivists often will propagate it.
a prefix name such as kilo- should be pronounced and stressed exactly the same regardless of the unit it is combined with. For example in
- kilo- watt'
- kilo -gram'
- kilo- metre'
With regard to this last unit (the kilometre), there is some controversy over the "correct" pronunciation. Consistency suggests that the word kilometre' should be pronounced with stress on the first syllable (KIL-o-metre). However, it is recognised that some people stress the word on the second syllable - thus, "kilOMetre". It is likely that this pronunciation is based on the analogy with "barometer" and "speedometer". However, the latter are obviously measuring instruments - not units of measurement. It may be that the confusion first arose from the American spelling of "metre" as "meter".
Whatever its origins, the practice of stressing the second syllable is widespread, especially in the USA, Australia and Ireland. Languages are not necessarily logical, and it is more important that people should use kilometres than that they should pronounce the word in a particular way.
Having said this, UKMA believes in being consistent and recommends following the simple rules outlined above - that is, the stress should be on the first syllable. Indeed, it would be appropriate if this rule were to be taught in schools as part of the national curriculum.
The OED says
The stress is marked by Webster (1828), Craig, and Cassell as kiˈlometre.
I don't know if this is supposed to indicate that this is the oldest attested stress pattern.