Your subject line has a question in it, but the body does not. Is your question "Where are scripts executed?" If that is your question then yes, you're correct, it should be the home directory of the user you logged into the remote server with. You can see by doing something like
ansible servera -m shell -a "ls"-u username
This will print the contents of the working directory when a user logs in. Try it with the become options to see what the result is. You can use an absolute path to call a script. Or use the "chdir" option for shell, something like
---
- hosts: collector
become: true
become_method: su
become_user: root
tasks:
- name: stop eum
shell: bin/eum.sh stop
args:
chdir: /opt/AppDynamics/EUEM/eum-processor/
should guarantee that you're in the correct location.