All kinds of organizations and people build infrastructure; churches and companies also promulgate laws and provide dispute-resolution forums. What distinguishes the state from other organizations that perform similar tasks is that the state holds a monopoly on (legitimate) violence. This is not a "narrow and immature" point of view; this is the normal definition. Quoting Wikipedia:
The most commonly used definition is Max Weber's,[6][7][8][9][10] which describes the state as a compulsory political organization with a centralized government that maintains a monopoly of the legitimate use of force within a certain territory.
Churches and companies don't promulgate "laws", they promulgate "rules." Rules have consequences to breaking them, but these consequences may vary widely in effect on the individual. A priest can try to make you feel guilty for breaking the religious edicts of his church, but if you don't consider his self-proclaimed authority in your life to have any basis, then guilt will be an ineffective consequence for breaking the rule. Companies, similarly, can fire you for breaking their rules, and while this deprives you of money and possibly opportunities, it does not deprive you of your life or freedom.
In short, the authority of these private organizations are opt-out, while the authority of the state is not.
It's circular logic to say that a state has a monopoly on only legitimate violence since it's the same state which defines which violence is legitimate and which is not. Can you think of any organizations that successfully exercise a monopoly on illegitimate violence? Successful states do not allow such activities to continue for long.
A state is, as others have already said, simply a political institution which exercises a monopoly on violence and determines how to apply it within its political domain.
It sounds like you think you disagree with me but I can't tell why. (Except for the semantic hairsplitting at the beginning, which is simply mistaken, but unrelated to the rest of the discussion.)
Perhaps a good intuition pump is how we react when this breaks down. If some group within a state's borders is using violence (i.e. sending soldiers to attack skyscrapers in another country), the attacked country has a cassus belli against the harboring country.
This feature is so critical to how the modern system treats territorial integrity that we sometimes refer to states which can't control violent groups within their own borders as "failed states."
Was this example really necessary? You basically said: If terrorists (who, by the way, happened to travel to Afghanistan once) attacked the WTC, the US have a casus belli against Afghanistan.
See, the reason why the war in Afghanistan is controversial is precisely because it is not clear that a country has a casus belli against another country if there is a (remote) connection of offenders to that country. In fact, the pilots lived and planned most of the plot in Germany.
A better example of a failed state would be the results of the war on drugs in Mexico.
The most commonly used definition is Max Weber's,[6][7][8][9][10] which describes the state as a compulsory political organization with a centralized government that maintains a monopoly of the legitimate use of force within a certain territory.