Exotic definition of fork

[[Le14]](https://arxiv.org/abs/1612.09375) defines fork like this: <img width="500" alt="Image" src="https://github.com/user-attachments/assets/3a3f53b5-a48f-400b-a783-d9080a105e28" /> nLab defines fork like this: <img width="465" alt="Image" src="https://github.com/user-attachments/assets/fc0e1dd2-0530-4079-94e2-5850e26ebc9f" /> (https://ncatlab.org/nlab/show/fork) According to their definition, a fork is an instance of cone. In Napkin, it’s very similar, but the $X \rightrightarrows Y$ part is defined as a fork, and a cone built over $X \rightrightarrows Y$ is called a “cone over this fork”: <img width="600" alt="Image" src="https://github.com/user-attachments/assets/1527a5f5-ab09-4fa7-ad90-c5139163e6d6" /> In other words, what [Le14] and nLab call a “fork,” we refer to as a “cone over this fork.” Is there a particular reason for this difference in definitions?