Use of Azure DevOps pipeline completion trigger when triggering pipeline has a failure

I was able to manage my requirement through the pipeline completion trigger itself. It is possible if we define stages in the triggering pipeline. I'm posting the answer if someone else looking for the same approach.

1. Need to define the triggering pipeline definition with stages. Also, we need to make sure that at least one stage should be successful every time. I already have few stages defined and hence this is totally matching with my requirement.

   Triggering pipeline YAML definition: (pipeline name: pipeline1)

   trigger: none
   pr: none
   
   pool:
     vmImage: 'ubuntu-latest'
   
   stages:
     - stage: stage_1
       displayName: Stage-1
       jobs:
         - job: greeting
           displayName: Greeting
           steps:
             - script: |
                 echo "Hello world!"
                 exit 1
   
     - stage: stage_2
       displayName: Stage-2
       condition: always()
       jobs:
         - job: thanking
           displayName: Thanking
           steps:
             - script: |
                 echo "Thank you!"
   

2. Define the pipeline completion trigger with stage filters for the triggered pipeline.

   Triggered pipeline YAML definition:

   trigger: none
   pr: none
   
   resources:
     pipelines:
       - pipeline: Pipeline_1
         source: pipeline1
         trigger:
           stages:
             - stage_2
   
   pool:
     vmImage: 'ubuntu-latest'
   
   jobs:
     - job: greeting
       steps:
         - script: |
             echo "Hello world!"
   

Then the triggered pipeline will be triggered irrespective to the stage_1 in the triggering pipeline since stage_2 will be kept successful in each run.