Make design note pin badge compliant with Pajamas

What does this MR do and why?

#344297 (closed) & #344294 (closed)

This MR converts badge badge-pill present in ~/design_management/components/design_note_pin.vue to use a new class .design-note-pin and make it compliant with Pajamas Design System.

Screenshots or screen recordings

The new pin looks identical to the old ones, without the .badge .badge-pill classes.

How to set up and validate locally

Numbered steps to set up and validate the change are strongly suggested.

MR acceptance checklist

This checklist encourages us to confirm any changes have been analyzed to reduce risks in quality, performance, reliability, security, and maintainability.

• I have evaluated the MR acceptance checklist for this MR.