That's a website of the Swiss government calling its system of governance a direct democracy. I'm talking about political theory, not the Swiss government's publications about itself nor the Swiss curriculum for "Staatskunde" lessons in schools.
I understand that Switzerland calls itself a direct democracy. I understand that many other representative democracies refer to the Swiss form of governance that way. I understand that this language may even have ended up in laws and legal interpretations, but that doesn't change that "direct democracy" means something very specific (i.e. literally everyone voting on every issue, often in the form of proxy voting or delegation) in political theory. I'm not saying Switzerland does not offer more voter participation than other representative democracies. It is a lot closer to direct democracy than any of its neighbors or fellow European democracies. But it's not a direct democracy.
Think of direct democracy like functional programming: there are plenty of programming languages that were originally conceived of as procedural or object-oriented or class-based that claim to support "functional programming" and some are actually very good at it but that does not make them functional programming languages in the same way as languages designed as functional programming languages from the start. Switzerland started out as a representative democracy and introduced and strengthened its participatory instruments over time to behave more like a direct democracy (check out the PDF at the end of the page you linked, it even uses the same narrative of Athens and the French revolution used by other representative democracies to explain their origins). That does not make it a direct democracy, it just makes it a representative democracy with exceptionally good participatory instruments.
It makes perfect sense for Switzerland to describe itself as a direct democracy when it wants to contrast itself with all the other representative democracies because those instruments borrowed from direct democracy set it apart. But that's like Ruby bragging about being a functional programming language by comparing itself to Python and Java because Haskell and Lisp are not part of the conversation.
These may all be interesting trains of thought, but they have little to do with the state treaties and processes at hand, which are based on agreed terminology (insofar as human language allows such a thing at all) and which are the reference for the assessment of the case at hand. If you are interested in the subject, you should study the works of established state theorists. Wikipedia is a rather dubious source for a scientific discussion of a topic.
State theory and specifically the state theory relating to legal practice like "state treaties and processes" is a very small subset of political theory, let alone political philosophy. There is a broad range of political theory outside that narrow subset and "direct democracy" has a specific meaning distinct from representative democracy.
Redefining it as a subset of representative democracy through the introduction of participatory processes makes sense when talking in a context where representative democracies are taken for granted but it is extremely unhelpful when talking in a broader scope where representative democracies are merely one of many categories of systems. There's a reason the PDF references the French Revolution rather than the Paris Commune and describes democracy merely in a (largely ahistorical TBH) European context informed by the Enlightenment era and limits the idea of decentralisation to the federalism of cantons rather than exploring it further: it's not about analysis, it's about creating a national mythology and a historical narrative that has the present system as an inevitable end state that can only be built upon but not fundamentally changed.
And all of this is really a red herring because the original argument was about whether the system Switzerland has makes it unnecessary to appeal to a super-national court in order to force a decision. And given that referendums and public initiatives still require a majority in order to pass the answer is clearly no.
Given that the claim is that the (in)action of the Swiss government accelerates/intensifies the climate catastrophe and that this effectively violates the human rights of a minority group (i.e. seniors are more likely to die from severe weather events like heatwaves) and that referendums have not helped change the (in)action of the government, a referendum is the wrong instrument as the question is not what the majority wants but whether a minority has their fundamental rights violated.
You can argue that the ECHR court was wrong to agree that this is a violation of the rights asserted by the ECHR. You can argue that Switzerland should leave the ECHR and replace it with a different human rights bill - and assuming you have the requisite legal status in Switzerland you can even attempt to force this question with a public initiative. You could also simply argue that they should have sued the Swiss government in a Swiss court of law first (I'm not sure how the Federal Supreme Court works but presumably the process would be a bit more involved) before taking this to an international level. But the ECHR court is ultimately the right addressee when arguing a signatory's violation of the ECHR, "direct" democracy or not.
I understand that Switzerland calls itself a direct democracy. I understand that many other representative democracies refer to the Swiss form of governance that way. I understand that this language may even have ended up in laws and legal interpretations, but that doesn't change that "direct democracy" means something very specific (i.e. literally everyone voting on every issue, often in the form of proxy voting or delegation) in political theory. I'm not saying Switzerland does not offer more voter participation than other representative democracies. It is a lot closer to direct democracy than any of its neighbors or fellow European democracies. But it's not a direct democracy.
Think of direct democracy like functional programming: there are plenty of programming languages that were originally conceived of as procedural or object-oriented or class-based that claim to support "functional programming" and some are actually very good at it but that does not make them functional programming languages in the same way as languages designed as functional programming languages from the start. Switzerland started out as a representative democracy and introduced and strengthened its participatory instruments over time to behave more like a direct democracy (check out the PDF at the end of the page you linked, it even uses the same narrative of Athens and the French revolution used by other representative democracies to explain their origins). That does not make it a direct democracy, it just makes it a representative democracy with exceptionally good participatory instruments.
It makes perfect sense for Switzerland to describe itself as a direct democracy when it wants to contrast itself with all the other representative democracies because those instruments borrowed from direct democracy set it apart. But that's like Ruby bragging about being a functional programming language by comparing itself to Python and Java because Haskell and Lisp are not part of the conversation.