(build) Make workflow similar to Cake.Discord

There are still some build errors, so moving the workflow to be similar
to what is in Cake.Discord, since this was updated recently by Nils,
and is known to be working.

.github/workflows/build.yml

@@ -19,7 +19,7 @@ jobs:
     runs-on: ${{ matrix.os }}
     strategy:
       matrix:
-        os: [windows-latest, ubuntu-18.04, macos-13]
+        os: [windows-2022, ubuntu-22.04, macos-12]
 
     env:
       AZURE_PASSWORD: ${{ secrets.AZURE_PASSWORD }}
@@ -51,7 +51,18 @@ jobs:
         uses: actions/cache@v3
         with:
           path: tools
-          key: ${{ runner.os }}-tools-${{ hashFiles('recipe.cake') }}
+          key: ${{ runner.os }}-tools-${{ hashFiles('recipe.cake', '.config/dotnet-tools.json') }}
+
+      - name: Setup required dotnet versions
+        uses: actions/setup-dotnet@6bd8b7f7774af54e05809fcc5431931b3eb1ddee # v4.0.1
+        with:
+          dotnet-version: |
+            2.1.818
+            3.1.x
+            5.0.x
+            6.0.x
+            7.0.x
+            8.0.x
 
       - name: Build project
         uses: cake-build/cake-action@v1