From the Apache 6 documentation: https://tomcat.apache.org/tomcat-6.0-doc/config/context.html in the Attributes section of Context, the documentation for the path attribute specifies:
This attribute must only be used when statically defining a Context in server.xml. In all other circumstances, the path will be inferred from the filenames used for either the .xml context file or the docBase.
Even when statically defining a Context in server.xml, this attribute must not be set unless either the docBase is not located under the Host’s appBase or both deployOnStartup and autoDeploy are false. If this rule is not followed, double deployment is likely to result.
The same documentation exists in Tomcat 7, so I tried the following on Tomcat 7 and I managed to deploy only the manager application.