meta: Transition the SDK to use issue types over issue type labels

### Description

We are using issue types as per the beta: https://github.com/orgs/community/discussions/112806

Let's rely on these instead of using the `Type: bug` and `Type: improvement` labels


```[tasklist]
### Tasks
- [ ] https://github.com/getsentry/sentry-javascript/pull/13211
- [x] Categorize everything with issue types
- [x] Remove old issue type labels from bug template
```

