Extraterritorial really just is an adjective that means "outside the territory". Which has two different and opposed, though sometimes related, uses in legal parlance.
When a country exercises its authority outside its territory, it creates an
extraterritorial jurisdiction. When a country treats a part of its territory or a person present on its territory as if it wasn't inside its territory, it grants an
extraterritorial privilege. Originally, the latter case was also called "exterritoriality" in the past, but the term fell into disuse.
It is a common misconception that foreign embassies and consulates have extraterritorial privilege. The diplomatic personnel have diplomatic immunity, while the diplomatic missions and vehicles have diplomatic inviolability. That is, the person cannot be arrested or imprisoned, the places and vehicles cannot be searched.
In this day and age, it is not so rare for democratic countries to waive their diplomats' immunity for cases that are not related to their duties, like car accidents.
Diplomatic immunity can practically amount to an extraterritorial privilege
in a given situation. When the local authorities don't even start a legal procedure because they know immunity or inviolability will get in their way, they effectively grant an extraterritorial privilege. It is not what the Vienna Convention ask for, but this what the results may be.
Foreign governments and the convention don't require it for a reason: the extraterritorial privilege is a double-edged sword. If you punch someone in the face inside the country, you cannot be charged because you weren't there. But if you get punched, the other person also cannot be charged because you weren't there.
The extraterritorial privilege still exists in some places though: the facilities belonging to international organizations are often granted it. That's the case for the UN or NATO headquarters for instance, or some buildings that belongs to the Catholic Church inside Rome.
Why not just resort to diplomatic immunity and inviolability? For NATO headquarters, it's because military personnel involved in the planning and execution of war operations don't qualify as diplomatic personnel. For the UN, it was to account for diplomats from countries which have no diplomatic relationships with or aren't recognize by the United States.
What prevent these places to be plagued with crime is extraterritoriality... the other one, the extraterritorial jurisdiction. Historically, your typical case is piracy inside international waters. A ship inside international waters is considered an extraterritorial jurisdiction of the country it carries the flag, so pirate crew boarding it can be charged and prosecuted there.
When an US Army officer is charged for murder committed in Iraq or inside the NATO headquarters in Belgium, it establishes an extraterritorial jurisdiction. A lot of countries actually allow themselves to try their citizens for crime they committed abroad. Some also allow themselves to try foreigners for crime they committed against one of their citizens abroad. And there are more disputed cases, like the US charging a Swiss company for dealing with the Iranian government that physically took place in Zurich and Tehran.
So, yes a place can have the extraterritorial privilege and also be an extraterritorial jurisdiction at the same time, and from the same country (in the case of a crime committed inside the UN headquarters for instance, it is considered as being outside US territory, but subject to US laws).
Extraterritorial jurisdiction are considered on a case by case basis. If you take a look at those "anecdotal evidences" that jurists usually call "precedents", the Fourth Circuit ruled that an US embassy is an extraterritorial jurisdiction for manslaughter in
United States versus Erdos, but the Supreme Court ruled that an US embassy was not for a civilian lawsuit against a foreign state in
Persinger versus Islamic Republic of Iran.
An US embassy is not actual US territory, it is a "special territorial jurisdiction of the United States". Some US laws apply, some don't, some do differently (in US v. Erdos, much of the debate was about where the trial should take place for instance). But this is because the US legal systems allow extraterritorial jurisdiction. The Russian may have a different opinion on what happens inside their embassies. Depending on the local practice of law, some other countries may rule extraterritorial jurisdiction as legally impossible, and their embassies thus effectively are lawless zones.
More important, things may be different as well in Shadowrun, which as a game tend to be closer to "Hollywood reality" than "legal reality". The foreign diplomat getting away with murder is a staple afterall.
