Environment detail:
Orchestration designer: 7.2.3
Tomcat 9.0.8
JDK: 1.8
Eclipse (Photon) 4.8

Is there a configuration on OD or Tomcat that tells where the application is deployed on Tomcat?

This is a new environment so I'm not sure if the configuration is complete/accurate.

In any event, when I deploy a war file, the content path show the local path of the war file (C:/UsersBRIAN.NESBITTwar filesAnimalControl_v1) instead of /AnimalControl_v1) as it should be. See attachment. I used the lower deploy war file option. The upper option does not work at all.

On Tomcat Web Application Manager, check under columns 'Running' and 'Commands' of each App.

If you deployed the WAR file correctly, 'Running' will be 'true' and you 'll be able, under column 'Commands', to Stop, Reaload and Undeploy the App.

Please, take a look at the first Attachment.

Name under column 'Path' has the format /Application_name so "/AnimalControl_v1" is correct.

One question: did you exported on OD and deployed on Tomcat, runtimeconfig.war and runtimeSupportTomcat9.zip ?
They are both necessary if you want your App properly run on Tomcat.
I don't see "/runtimaconfig" on your Tomcat,it should be the last App after "/manager", take a look at the second Attachment. You should have done that the first time you used OD or after the last OD update.
If you don't know how to do, just reply here and I can provide step-by-step guide for OD 7.2.3.

Thanks! A step-by-step guide would be great!

Sorry for delay. Here you go:

https://www.devconnectprogram.com/fileMedia/download/0d1b3002-5f32-4820-9d65-7a6b347dbe30

Take a look at section 2.1 ' Export and install Runtime files '.

Thank you!