Update Nuxt to v3.13.1

[obulat]

Fixes

Description

This PR re-adds the changes for updating Nuxt to v3.13.1. I removed the changes related to i18n for easier testing.

Vue also had to be updated to the one matching this Nuxt version.
The patch release fixed the serverDir in nuxt.config, so I reverted the changes to folders for now.
I also added the recommended workaround for vue-router types.

Testing Instructions

The CI should pass, and the app should work as expected. Since the server folder was moved, it is important to check that the server routes work as usual (/healthcheck, /robots)

Checklist

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[obulat]

Drafting because there are typing problems that we've previously seen and couldn't identify the origin: the component doesn't have $route property in this PR

[zackkrida]

@obulat just curious why the pinia reversion was necessary. This looks good though

[obulat]

@obulat just curious why the pinia reversion was necessary. This looks good though

Just to minimize the number of changes in one PR, to make reverting easier if necessary

[zackkrida]

LGTM. I appreciate the constrained scope.