The original definition of DevOps was what you describe in your second paragraph - development teams being responsible for operations.
Now DevOps means "We've created a DevOps team! They do DevOps!" Frequently its the managers of old school operation teams changing their names to DevOps teams to get a promotion.
"Giving developers operational responsibilities has greatly enhanced the quality of the services, both from a customer and a technology point of view. The traditional model is that you take your software to the wall that separates development and operations, and throw it over and then forget about it. Not at Amazon. You build it, you run it." - Werner Vogels, Amazon CTO (2006)
DevOps meant that Developers are also responsible for Operations. But that meaning is lost to time, just like hacking meaning programming.
It may seem pedantic, but it's an important difference: that's _not_ what the original definition was. Instead, it was a cross-functional team of developers and operators being responsible for a whole system.
Yes, I should not have said a "development team", as it makes a distinction between "people who create new software" and "people who run that software in production environment". Your definition is better.
Now DevOps means "We've created a DevOps team! They do DevOps!" Frequently its the managers of old school operation teams changing their names to DevOps teams to get a promotion.