I believe our politics could possibly be decentralised because of how divorced party and government are here, due to the separation of executive and legislative functions. If the executive and legislature are tied to eachother, it becomes very easy to tell who the leaders are. But when you have three separate, independent yet interdependent chambers of power, leadership can often be divided.
Which I am not sure is a good thing in a modern democracy.
Oh yes, this is why I always found European critiques of the electoral college silly, considering that at the end of the day, the functional leader of nations on both sides of the pond is indirectly elected.
Well, I just think it's redundant, but it's really not that of an issue.
You're confusing that with the Putin gambit in Russia
Guilty

No, really, in most countries whose political systems I've studied, the limits only prevent too many consecutive terms in office, so I assumed it was the same in the US. Just proves how dangerous such assumptions are.
Winner has already pointed some things out, but if you really want to know how most presidents of European states work, think of the Queen without having their faces on stamps and (almost) without the extra wasted taxpayer money.
Our president
a) has his face on stamps
b) lives in the Prague Castle, the traditional seat of kings
c) has his own coat of arms, fanfare, and seal
d) has a ceremonial castle guard at his disposal
(e) is an idiot)
It's so close to a monarchy that some foreign visitors think he's a king. He certainly likes to act as if he was
That's not technically true. The parties do have set leaders, but they serve a more behind the scenes role, organizing campaigns and such (national committees) or actually legislating (House/Senate majority/minority leaders). The most senior guy usually becomes the de facto spokesman. For the party occupying the White House, it's the President; for the other party, it's the leader of whatever chamber they happen to control (currently the Speaker of the House). It's a little distributed, but right now, Obama is Mr. Democrat, and Boehner is Mr. Republican.
The chaos comes in for the challenging party come election time, as they have a bunch of voices competing to be the presidential nominee, who will then by virtue of the hierarchy leap-frog the legislative leaders and become the new party spokesman.
What I mean is that American political parties aren't really parties if we compare them with the European concept of parties. They're more like a common ideological platform, a coalition of like-minded groups. In Europe, parties usually have clear, long-standing political programmes and are ideologically homogeneous (there are of course internal debates and struggles, but on much smaller scale). Usually they are also very centralized - there is a chairman who is in most cases also the leading candidate for premiership, and he has a great deal of control over the party's policy. Members of Parliament elected for a party are expected to obey the will of the party's leadership. A backbencher revolt has the potential to bring a government down quickly because ruling cabinets in parliamentary systems rest on a parliamentary majority. When they lose it, they usually fall, unless they strike some sort of a deal with the opposition.
To us, the American system really
looks pretty chaotic. First, the MPs (congressmen/women + senators) are much less loyal to the leader of their party, or their party in general. Second, since the executive branch doesn't rely on a parliamentary majority, it common that the president rules even though the Congress is dominated by his opponents. Third, political programmes in America are very candidate-centric; for example, I haven't noticed any "Republican political programme", only the programmes of prominent Republicans. Fourth, the elections are held too often.
It's a small wonder you've ever managed to have a stable government
Exactly. It's not as simple as Winner made it out to be over here as well - for example, the Social Democrats party (which is likely to replace the current government according to current polls) doesn't know whom to make their candidate for chancellorship yet - and their chairman isn't very likely to become it. They're even thinking about holding US-style primaries.
I only explained how it
usually works. In my opinion, the concept of split leadership (when the premier/chancellor/prime minister isn't the leader of his party) is an unstable one. The head of the government is then
weaker than the leader of his party, which is a potential source of instability and confusion. In this country, this situation has only occurred when the ruling party elected a new chairman while the old one was still the premier - they chose to leave him in office in order to avoid the whole process of cabinet reshuffle followed by a constitutionally required vote of confidence in the parliament. Each time it was clear the premier lost a great deal of credibility and power.
Isn't the reason why SDP wants someone else than the party chairman for chancellorship the fact that he isn't very... charismatic?