This seems like needless pedantry at least for a conversation outside of school. The USSR (with whom the Cold War was being fought) was dissolved and Germany was reunified. For all intents and purposes it did end, just a new conflict (now with Russia) began.
Seeing as the Cold War was a framework for conflicts, not a conflict in itself, I find it extremely illuminating to consider the view of the opposing party. Not a technicality in the slightest.
Andrei Piontkovskiy, in addition to two known World Wars, considers the Cold War as the World War III and the current war in Eastern Europe as the World War IV. His parallels are that WWII was fought by Germany dissatisfied by the results of WWI, and WWIV is fought by Russia dissatisfied by the results of WWIII.
World wars involve many countries, and Cold War definitely qualify. Today's war is a pretty active, quite large "hot" war, which also involves many countries - even though most fight by proxy.
I also think "Cold War II" for the current situation is more fitting.
I think if there is any useful distinction between "hot" and "cold" world wars then it's most likely whether super powers are in direct military conflict with each other or whether military confrontation is "only" through proxy wars.
Note that the original cold war wasn't very "cold" for much of the world either - the only thing that didn't happen was direct millitary confrontation between the US and USSR. Nevertheless there were lots of local conflicts and proxy wars where each bloc was backing a faction.
In the today's war in Ukraine one country - Russia - fights directly, not from proxies, and the other side - mostly USA, but also other Western countries - supply weapons, volunteers, intelligence services, training. It is comparable with Vietnam war, right, but not already with Afghan war of 1980-s, or small conflicts around the world. The scale of war is also quite large, the level of directly fighting forces is much more comparable.
Should we admit that world wars don't need to involve superpowers - or at least only superpowers? The term wasn't that much applicable before end of WWII.
So here we can argue that in WWIV a non-superpower fights - Ukraine, on its territory, a superpower - USA, merely - but with principal results - supporting Ukraine, and the rest of the West. We may not call it a proxy war - I agree, it's a rather poor comparison - but for WWIV term it is another matter.
World wars don't need to involve superpowers, but they need to involve large part of the world. Russian invasion on Ukraine doesn't, and it's unlikely to escalate - Russia can't, because they have neither people, hardware, or allies, and the defending countries don't have a reason to.
From another perspective all of it is a continuation of the Great Game, the Anglosphere/Russia conflict dating back to the 1800s that never really stopped, and was merely put on pause for a few years a couple times (mostly when the Anglos felt that other continental Europeans were consolidating enough power to be an even greater threat than Russia.)
You make it sound like NATO was unilaterally pushing for this. Eastern European countries were begging to join NATO. All of them had been independent multiple times over the centuries, always ending up under Russian control. NATO offered a plausible mechanism to end the historical cycle--an historical cycle for which Russia, in 2022, is proudly nostalgic and not afraid to go to war to continue.
Moreover, national security is expensive, especially for small countries who cannot benefit from scale--they need to spend much more for even minimal deterrence. For newly independent nations, NATO provides leverage for their security expenditures. More importantly, it also motivates peaceful resolution of conflict among neighboring NATO states, which makes NATO a keystone institution for peace in Europe, Russia notwithstanding.
What British and France interests were served when they agreed to sign an alliance with Poland and even went to war with Germany over it (eventually)? It's the same basically, except that this time, I hope, the alliance is more credible and Russia is not willing to test it.
There are plenty of economical and political reasons due to which is it beneficial to keep Latvia and by extension the entirety of Eastern Europe outside of direct or indirect Russian control.
Ukraine is a pretty good example of a country which was mostly ignored both by NATO and the EU so as not to antagonize Russia. It remained a failed state until 2014 and I assume we all know what happened afterwards.
