this is completely dependent on art style. Saying "Anime" style doesn't mean much. Noses work great on 80s or 90s style. Noses don't look so good on Chibi style. Which is what this picture is.
That being said, noses are usually drawn with omissions due to the flatness of the style and Ukiyo-e influence.
Here's Sadamoto's art for Wings of Honneamise, and even though there's detailed noses here, there's a few small signs of stylistic omissions already. (His later art is pretty normal in terms of noses, even though he's partially responsible for that) Modern anime with noses would be... Attack on Titan, I guess.
I like to point out that the nose on the left example would look a lot less... weird if the artist uses a darker tone of the skin color instead of just black. Basically it needs to be subtle if you don't want it to stand out too much, especially on a style as simplified as this.
