That is precisely what they mean.
That’s why it’s annoying to hear someone say something like: “We’re anxious to get started on this wonderful project” when they are clearly looking forward to it rather than dreading it.
Yeah. You can be both eager and anxious at the same time but they’re still conflicting emotions.
In fact you need two distinct words to convey the conflict of emotions sometimes. I’m sure astronauts on the launchpad are often both eager and anxious.
I believe the words anxious and anxiety are closely related.
I have been both anxious AND eager to get started on some projects when I’m concerned about having enough time to finish something I want to work on and do a good job.