[ci] revise wasm CI (microsoft#24825)

### Description
<!-- Describe your changes. -->

revise WASM CI to run test as later step than publishing artifacts. This
allows download the binary to diagnose test failures.


### Motivation and Context
<!-- - Why is this change required? What problem does it solve?
- If it fixes an open issue, please link to the issue here. -->

Author: fs-eire

.github/workflows/linux-wasm-ci-build-and-test-workflow.yml

@@ -65,27 +65,6 @@ jobs:
             --skip_tests
         working-directory: ${{ github.workspace }}
 
-      - name: Test (Node.js) (simd + threads)
-        # onnxruntime_test_all is currently only supported in Debug build because it requires exception, which is disabled in Release build.
-        if: ${{ inputs.build_config == 'Debug' }}
-        run: |
-          python ./tools/ci_build/build.py \
-            ${{ env.common_build_args }} \
-            --build_dir ${{ github.workspace }}/build/wasm_inferencing \
-            --test
-        working-directory: ${{ github.workspace }}
-
-      - name: Test (browser) (simd + threads)
-        # onnxruntime_test_all is currently only supported in Debug build because it requires exception, which is disabled in Release build.
-        if: ${{ inputs.build_config == 'Debug' }}
-        run: |
-          python ./tools/ci_build/build.py \
-            ${{ env.common_build_args }} \
-            --build_dir ${{ github.workspace }}/build/wasm_inferencing \
-            --wasm_run_tests_in_browser \
-            --test
-        working-directory: ${{ github.workspace }}
-
       - name: Build (simd + threads + JSEP)
         if: ${{ inputs.build_jsep == true }}
         run: |
@@ -143,6 +122,28 @@ jobs:
           name: ${{ inputs.build_config }}_wasm_webgpu
           path: ${{ github.workspace }}/artifacts/wasm_webgpu
 
+      - name: Test (Node.js) (simd + threads)
+        # onnxruntime_test_all is currently only supported in Debug build because it requires exception, which is disabled in Release build.
+        if: ${{ inputs.build_config == 'Debug' }}
+        run: |
+          python ./tools/ci_build/build.py \
+            ${{ env.common_build_args }} \
+            --build_dir ${{ github.workspace }}/build/wasm_inferencing \
+            --test
+        working-directory: ${{ github.workspace }}
+
+      - name: Test (browser) (simd + threads)
+        # onnxruntime_test_all is currently only supported in Debug build because it requires exception, which is disabled in Release build.
+        if: ${{ inputs.build_config == 'Debug' }}
+        run: |
+          python ./tools/ci_build/build.py \
+            ${{ env.common_build_args }} \
+            --build_dir ${{ github.workspace }}/build/wasm_inferencing \
+            --wasm_run_tests_in_browser \
+            --target onnxruntime_test_all \
+            --update --build --test
+        working-directory: ${{ github.workspace }}
+
       - name: Publish test results
         if: ${{ always() && inputs.build_config == 'Debug' }}
         uses: actions/upload-artifact@v4