Bump dependencies to latest, hold ESLint

This commit updates the project's npm dependencies to their
latest versions.

Updates to the following dependencies are on hold due to compatibility
issues:

- `@typescript-eslint/eslint-plugin`:
  - Blocked by `@vue/eslint-config-airbnb-with-typescript`
    (vuejs/eslint-config-airbnb#63).
- `@typescript-eslint/parser`:
  - Blocked by `@vue/eslint-config-airbnb-with-typescript`
    (vuejs/eslint-config-airbnb#63).
- `@vue/eslint-config-typescript`:
  - Blocked by `@vue/eslint-config-airbnb-with-typescript`
    (vuejs/eslint-config-airbnb#63).
- `eslint`:
  - Blocked by `@vue/eslint-config-airbnb-with-typescript`
    (vuejs/eslint-config-airbnb#65).
  - Blocked by `@typescript-eslint/eslint-plugin` and
    `@typescript-eslint/parser`
    (typescript-eslint/typescript-eslint#8211).

These dependencies remain at their current major versions, and
their status is documented in the `package.json` to inform future
updates.

This commit moves also `@types/markdown-it` to `devDependencies` which
was incorrectly included in `dependencies`.