Split ondevice training cpu packaging pipeline to a separated pipeline (#21485)

### Description
Right now our "Zip-Nuget-Java-Nodejs Packaging Pipeline" is too big.
This OnDevice training part is independent of the others, so it can be
split out. Then our NPM Packaging pipeline will not depends on this
training stuff.

### Motivation and Context
Similar to #21235 

Also, this PR fixed a problem that: "NuGet_Test_Linux_Training_CPU" job
downloads artifacts from "onnxruntime-linux-x64" for getting customop
shared libs, but the job forget to declare it depends on the
"Linux_C_API_Packaging_CPU_x64" which produces the artifact. Such
problems can be hard to find when a pipeline goes big.

tools/ci_build/github/azure-pipelines/c-api-noopenmp-packaging-pipelines.yml

@@ -112,17 +112,6 @@ stages:
     SpecificArtifact: ${{ parameters.SpecificArtifact }}
     BuildId: ${{ parameters.BuildId }}
 
-- template: templates/ondevice-training-cpu-packaging-pipeline.yml
-  parameters:
-    RunOnnxRuntimeTests: ${{ parameters.RunOnnxRuntimeTests }}
-    DoCompliance: ${{ parameters.DoCompliance }}
-    DoEsrp: ${{ parameters.DoEsrp }}
-    IsReleaseBuild: ${{ parameters.IsReleaseBuild }}
-    OrtNugetPackageId: 'Microsoft.ML.OnnxRuntime.Training'
-    AdditionalBuildFlags: '--enable_training_apis'
-    AdditionalWinBuildFlags: '--enable_onnx_tests --enable_wcos'
-    BuildVariant: 'default'
-
 - template: stages/java-cuda-packaging-stage.yml
   parameters:
     CudaVersion: 11.8

tools/ci_build/github/azure-pipelines/c-api-training-packaging-pipelines.yml

@@ -0,0 +1,51 @@
+parameters:
+- name: RunOnnxRuntimeTests
+  displayName: Run Tests?
+  type: boolean
+  default: true
+
+- name: DoCompliance
+  displayName: Run Compliance Tasks?
+  type: boolean
+  default: true
+
+- name: DoEsrp
+  displayName: Run code sign tasks? Must be true if you are doing an ONNX Runtime release
+  type: boolean
+  default: true
+
+- name: IsReleaseBuild
+  displayName: Is a release build? Set it to true if you are doing an ONNX Runtime release.
+  type: boolean
+  default: false
+- name: PreReleaseVersionSuffixString
+  displayName: Suffix added to pre-release package version. Only used if IsReleaseBuild is true. Denotes the type of pre-release package.
+  type: string
+  values:
+  - alpha
+  - beta
+  - rc
+  - none
+  default: none
+
+- name: PreReleaseVersionSuffixNumber
+  displayName: Number added to pre-release package version. Only used if IsReleaseBuild is true. Denotes the sequence of a pre-release package.
+  type: number
+  default: 0
+  
+stages:
+- template: stages/set_packaging_variables_stage.yml
+  parameters:
+    IsReleaseBuild: ${{ parameters.IsReleaseBuild }}
+    PreReleaseVersionSuffixString: ${{ parameters.PreReleaseVersionSuffixString }}
+    PreReleaseVersionSuffixNumber: ${{ parameters.PreReleaseVersionSuffixNumber }}
+- template: templates/ondevice-training-cpu-packaging-pipeline.yml
+  parameters:
+    RunOnnxRuntimeTests: ${{ parameters.RunOnnxRuntimeTests }}
+    DoCompliance: ${{ parameters.DoCompliance }}
+    DoEsrp: ${{ parameters.DoEsrp }}
+    IsReleaseBuild: ${{ parameters.IsReleaseBuild }}
+    OrtNugetPackageId: 'Microsoft.ML.OnnxRuntime.Training'
+    AdditionalBuildFlags: '--enable_training_apis'
+    AdditionalWinBuildFlags: '--enable_onnx_tests --enable_wcos'
+    BuildVariant: 'default'

tools/ci_build/github/azure-pipelines/templates/ondevice-training-cpu-packaging-pipeline.yml

@@ -317,3 +317,4 @@ stages:
     ArtifactSuffix: 'Training-CPU'
     StageSuffix: 'Training_CPU'
     NativePackagePrefix: 'onnxruntime-training'
+    CustomOpArtifactName: 'onnxruntime-training-linux-x64'