fix: broken links in READMEs and resolve link check issue (#21314)

* Update README.md

fix link for lightning-fabric in README

* fix broken links

* docs(README): add anchor link for "Why PyTorch Lightning?" section

---------

Co-authored-by: Bhimraj Yadav <[email protected]>
Co-authored-by: Shion Matsumoto <[email protected]>

Author: 3 people

README.md

@@ -64,9 +64,10 @@ Over 340,000 developers use [Lightning Cloud](https://lightning.ai/?utm_source=p
 - [Notebooks](https://lightning.ai/notebooks?utm_source=ptl_readme&utm_medium=referral&utm_campaign=ptl_readme): Persistent GPU workspaces where AI helps you code and analyze.
 - [Inference](https://lightning.ai/deploy?utm_source=ptl_readme&utm_medium=referral&utm_campaign=ptl_readme): Deploy models as inference APIs.   
 
+<a id="why-pytorch-lightning"></a>
 # Why PyTorch Lightning?   
 
-Training models in plain PyTorch is tedious and error-prone - you have to manually handle things like backprop, mixed precision, multi-GPU, and distributed training, often rewriting code for every new project. PyTorch Lightning organizes PyTorch code to automate those complexities so you can focus on your model and data, while keeping full control and scaling from CPU to multi-node without changing your core code. But if you want control of those things, you can still opt into [expert-level control](#lightning-fabric-expert-contro).   
+Training models in plain PyTorch is tedious and error-prone - you have to manually handle things like backprop, mixed precision, multi-GPU, and distributed training, often rewriting code for every new project. PyTorch Lightning organizes PyTorch code to automate those complexities so you can focus on your model and data, while keeping full control and scaling from CPU to multi-node without changing your core code. But if you want control of those things, you can still opt into [expert-level control](#lightning-fabric-expert-control).   
 
 Fun analogy: If PyTorch is Javascript, PyTorch Lightning is ReactJS or NextJS.
 
@@ -215,7 +216,7 @@ Explore various types of training possible with PyTorch Lightning. Pretrain and
 
 | Task | Description | Run |
 |------|--------------|-----|
-| [Hello world](#hello-simple-model) | Pretrain - Hello world example | <a target="_blank" href="https://lightning.ai/lightning-ai/studios/pytorch-lightning-hello-world?utm_source=ptl_readme&utm_medium=referral&utm_campaign=ptl_readme"><img src="https://pl-bolts-doc-images.s3.us-east-2.amazonaws.com/app-2/studio-badge.svg" alt="Open In Studio"/></a> |
+| [Hello world](https://lightning.ai/lightning-ai/studios/pytorch-lightning-hello-world?utm_source=ptl_readme&utm_medium=referral&utm_campaign=ptl_readme) | Pretrain - Hello world example | <a target="_blank" href="https://lightning.ai/lightning-ai/studios/pytorch-lightning-hello-world?utm_source=ptl_readme&utm_medium=referral&utm_campaign=ptl_readme"><img src="https://pl-bolts-doc-images.s3.us-east-2.amazonaws.com/app-2/studio-badge.svg" alt="Open In Studio"/></a> |
 | [Image classification](https://lightning.ai/lightning-ai/studios/image-classification-with-pytorch-lightning?utm_source=ptl_readme&utm_medium=referral&utm_campaign=ptl_readme) | Finetune - ResNet-34 model to classify images of cars | <a target="_blank" href="https://lightning.ai/lightning-ai/studios/image-classification-with-pytorch-lightning?utm_source=ptl_readme&utm_medium=referral&utm_campaign=ptl_readme"><img src="https://pl-bolts-doc-images.s3.us-east-2.amazonaws.com/app-2/studio-badge.svg" alt="Open In Studio"/></a> |
 | [Image segmentation](https://lightning.ai/lightning-ai/studios/image-segmentation-with-pytorch-lightning?utm_source=ptl_readme&utm_medium=referral&utm_campaign=ptl_readme) | Finetune - ResNet-50 model to segment images | <a target="_blank" href="https://lightning.ai/lightning-ai/studios/image-segmentation-with-pytorch-lightning?utm_source=ptl_readme&utm_medium=referral&utm_campaign=ptl_readme"><img src="https://pl-bolts-doc-images.s3.us-east-2.amazonaws.com/app-2/studio-badge.svg" alt="Open In Studio"/></a> |
 | [Object detection](https://lightning.ai/lightning-ai/studios/object-detection-with-pytorch-lightning?utm_source=ptl_readme&utm_medium=referral&utm_campaign=ptl_readme) | Finetune - Faster R-CNN model to detect objects | <a target="_blank" href="https://lightning.ai/lightning-ai/studios/object-detection-with-pytorch-lightning?utm_source=ptl_readme&utm_medium=referral&utm_campaign=ptl_readme"><img src="https://pl-bolts-doc-images.s3.us-east-2.amazonaws.com/app-2/studio-badge.svg" alt="Open In Studio"/></a> |

src/pytorch_lightning/README.md

@@ -9,7 +9,6 @@ ______________________________________________________________________
 
 <p align="center">
   <a href="https://www.pytorchlightning.ai/">Website</a> •
-  <a href="#key-features">Key Features</a> •
   <a href="#how-to-use">How To Use</a> •
   <a href="https://lightning.ai/docs/pytorch/stable/">Docs</a> •
   <a href="#examples">Examples</a> •