feat: Built with webstudio badge check on publish

[kof]

Description

Steps for reproduction

1. click button
2. expect xyz

Code Review

• hi @kof, I need you to do
  • conceptual review (architecture, feature-correctness)
  • detailed review (read every line)
  • test it on preview

Before requesting a review

• made a self-review
• added inline comments where things may be not obvious (the "why", not "what")

Before merging

• tested locally and on preview environment (preview dev login: 0000)
• updated test cases document
• added tests
• if any new env variables are added, added them to .env file