Personally, I use "American" in your intended way, mostly because it's been so damned long since you guys unilaterally took over the name that it hardly seems worthwhile fighting for it.
I'm frankly surprised that this it the first you've heard of it - I've seen the term "USian" used with no possible connotation of disrespect for years, often by Americans themselves. IMHO it's unbecoming that your first assumption upon seeing this word is to assume that he's hostile against your country.
The stereotype of the snobby European prick is about as tired as that of the ignorant American redneck.
Trust me on this one, if you've spent any time on the Internet you know it's a passive-aggressive ditch at the US (at least common enough to deserve a couple notes in Urban Dictionary). Anyway, when did it become the role of members outside a group to tell them whether or not they should be offended by the name they are called?
The problem is that the name "America" is hardly exclusive to the USA. It's as if a country called "The Grand Duchy of Europe" referred to its citizens as "Europeans", to the great dismay of everyone else in Europe.
This is completely irrelevant. Citizens of the USA have an official demonym in the English language and it is American. The argument that it is an attempt to avoid ambiguity is bullshit because no one ever uses the term Americans to refer to all inhabitants of the North and South Americas.
Um, when they started calling them names? Now, don't take this the wrong way, but that sounds a little defensive. It's quite normal for people to explain the reason they chose a particular characterization of a group, that name catching on, and the other group getting affronted at being called a group. At this point a new characterization, one without connotations, is found. All along, people keep explaining that there's nothing bad at being in a group, or alternately that grouping is only done to divide and conquer.
no one ever uses the term Americans to refer to all inhabitants of the North and South Americas.
And that'll be because they know they'd be misunderstood. I used to call people from the USA "people from the United States", but that got too cumbersome. So I switched to Yanks - fully aware that it is also a poor choice of word, but at least more specific and apolitical around here. I've most recently transitioned to "American" plus qualifiers (such as state), since I've been reading stuff written by themselves. I don't see any other words coming beyond the horizon, except maybe cultural ones.
However, I wouldn't put calling all inhabitants of the Americas "American" past biologists and other non-geopolitically minded people.
If the European Union ever turned into a real country rather than a loose confederation of allied states (just like the US did between 1776 and 1865), I bet you'd call citizens of that country "Europeans". Unless their respective countries had joined the EU in the meantime, the Norwegians, Swiss, Russians, Icelanders, Turks, Belorussians, Ukrainians, Moldovans, Serbs, Croats, Macedonians (or is that Former Yugoslav Republicans?), Kosovans, and Bosnians may or may not be annoyed at that usage.
 Some countries call Macedonia the "Former Yugoslav Republic of Macedonia", since some people in Greece are annoyed that they named their country after a region that extends into Greece.