The words "milli" and "micro" are not the problem itself. The problem is that this would introduce an additional, third unit to the two units we already have (satoshi & bitcoin).
I'd just like to layer my own observations atop this, being as unbiased as I can manage. I'm running with your implicit assumption that "bitcoin", "millibitcoin", and "microbitcoin" are distinct units.
The bitcoin is by far the most common unit in the Bitcoin system. This was introduced by satoshi with the release of the code (early 2009). It is the only unit sufficiently widely recognised that it has been entered into a number of reputable dictionaries (e.g. OED). The symbol "BTC" is the most popular, but
BTC and "XBT" are also frequently seen.
The millibitcoin is easily the second most widely used unit today. This follows naturally from "bitcoin" and the SI prefix "milli-" (1795) and was first discussed on the forum in early 2011. I doubt it's in any dictionaries but there is basically a consensus on the term. The symbol used is almost always "mBTC". Nicknames vary: "millibit", "mill", "millie".
The satoshi is the third most common unit. It was introduced in early 2011 by bitcointalk (then bitcoin forum) user ribuck (originally as a name for 0.000 001 BTC but this quickly changed to 0.000 000 01 BTC, currently the smallest possible unit). It is, as you observe, practically the only name for this unit. "SAT"/"Sat"/"sat" is relatively common as a symbol and a nickname.
The infamous 0.000 001 BTC unit is more the subject of flamewars than anything else, not least because two of the most popular names, "bit" and "microbit", are in direct logical conflict. It's possible both will become accepted (just as 1 calorie = 1 kilocalorie) but the issue is far from decided. I've seen many proposed names for this unit; off the top of my head (alphabetically, lower case): "bit", "centoshi", "fin", "finney", "hectoshi", "mic", "microbit", "microbitcoin", "mike", "mubit", "ubit", "xub", "zib". Selecting one name may well end up attracting negative attention to your platform so please tread carefully.
As I say, these are just my observations (reading something on Bitcoin roughly once a day for the past 4+ years). I'm happy to accept corrections and provide some citations on request.
This is by far the best post on this topic, in my opinion. Sums it up pretty much.
It also helps me to nail down the choices for my website:
I will certainly not want to risk choosing one of the many names for a 100-satoshi-sized unit, which leads me to avoiding this unit entirely.
I'll also avoid the mBTC unit for similiar reasons as franky1: The word "millibitcoin" implies that it is just a fraction of something, and the human brain doesn't like to deal with fractions — we prefer integers!
Of the remaining two units, BTC and SAT, the former is simply too large to be useful on my site.
Anyone who has to deal with bitcoin micro-transactions and is faced with the current situation will probably come to the same conclusion and quote values in satoshis.
This will be the interim solution as long as a consensus for naming the 100-satoshi unit is not found.
However, given that this discussion didn't make any tangible progress in the last six months, I'd bet that a word for 100 satoshis will not become universally adopted any time soon. Thus the interim solution might very well become the permanent solution.
Personally, I think that this interim solution, i.e. quoting values in satoshis, is in no way inferior to using a 100-satoshi-sized unit. Prices do not intrinsically need to have two decimal places on the right side, and instead could equally well be quoted as integers with no decimal point whatsoever.
I believe that the only reason why people want to have two decimal places to the right of the point is familiarity with their current local fiat currencies, which is — in the grand scheme of things — irrelevant.
Ofcourse one might argue that similiarity with the current system could ease understanding the new system.
However, while this approach works in general, it doesn't apply to currency units: I argue that teaching two currency units to newbies is more difficult than teaching only one unit,
even if the newbies are used to using two units in their local fiat currency. After all, whatever units you teach them are still
new to them, i.e. it's not the monetary units they already have.
Last but not least, one might rightfully complain that quoting values in satoshis will frequently incur large numbers. Transactions in the thousands will be the norm. Consequently, I guess, people will find various nicknames for thousand satoshis, analogous to the
grand in the dollar world.
