With the V1 search API you would geocode the city/district and if a polygon is available for it, a geometry ID will be included in the result. You can then use that with the Search Polygon API you linked to, to retrieve the polygon. You can try this out in this example: https://samples.azuremaps.com/?search=boundary&sample=search-for-boundaries For example, if you pass in "4ème arrondissement", one of the top results will be the "4th district" of the city of Paris.
With the V2 search services there is a new polygon service: https://learn.microsoft.com/en-us/rest/api/maps/search-v2/get-polygon?tabs=HTTP which takes in a latitude/longitude point or an address/place query. If an address/place is passed in, it is geocoded and the point for the first result is used. Then any polygon that intersects with the point that has the entity type you specified (https://learn.microsoft.com/en-us/rest/api/maps/search/get-search-address?tabs=HTTP#geographicentitytype). Districts typically are classified as "MunicipalitySubdivision" entity types.
The V1 service has a higher accuracy rate as you would be geocoding and selecting the result you want. Where as the V2 service is more flexible as you can retrieve all the boundaries for all entity types for a specific location which may be better suited for certain scenarios.