Germany, Austria and Switzerland are federal states like USA or Mexico too. This means, that all of the named countries are unions of a couple of former individual countries.
The USA is well know to be a federation of 50 states, and many people might also have learned that Mexico too is a federation that consists of 31 states.
- Germany is a federation ("Bund") of 16 states (singular: "Land" or "Bundesland", plural: "Länder" or "Bundesländer")
- Austria is a federation ("Bund") of 9 states (singular: "Land" or "Bundesland", plural: "Länder" or "Bundesländer")
- Switzerland is a federation ("Bund") of 26 states (singular: "Kanton", plural: "Kantone")
You see, that the Swiss federal states are not called "Länder" or "Bundesländer" but "Kantone", which makes it a little bit easier.
But also the federations themselves (Germany, Austria, Switzerland, USA and Mexico) are "Länder" (countries)
So, you have the situation, that in Germany and Austria the category-name is the same for the federal states as well as for the federation.
So, this all is correct:
Deutschland ist ein Land. - Germany is a country.
Das Saarland ist ein Land. - Saarland is a state.
Das Saarland ist ein Bundesland Deutschlands. - Saarland is a federal state of Germany.
Österreich ist ein Land. - Austria is a country.
Das Burgenland ist ein Land. - Burgenland is a state.
Das Burgenland ist ein Bundesland Österreichs. - Burgenland is a federal state of Austria.
And so, when you talk about a Länderspiel, then this is an international match (in sports).
A Ländervergleich is an international comparison, but is also might be a comparison between federal states within one federation.
But a Länderkammer always is a federal cabinet.
So, some terms beginning with Länder- always mean something that is international, some other always mean something that belongs to federal states of a federation, and some of these terms can mean both, depending on the context.
How can you know? Do the same that native speakers do: Learn it for each word, and if you are not sure, pay attention to the context.