How can I skip tests in maven install goal, while running them in maven test goal?

It sounds like you didn’t understand the concept of the build life-cycle in Maven. If you run mvn install all life-cycle phases (including the install phase itself) run before the install phase. This means running the following phases:

1. validate
2. initialize
3. generate-sources
4. process-sources
5. generate-resources
6. process-resources
7. compile
8. process-classes
9. generate-test-sources
10. process-test-sources
11. generate-test-resources
12. process-test-resources
13. test-compile
14. process-test-classes
15. test
16. prepare-package
17. package
18. pre-integration-test
19. integration-test
20. post-integration-test
21. verify
22. install

which means in other words the test as well as integration-test life-cycle phases are included. So without any supplemental information it’s not possible to change the behaviour as you wish it.

It could be achieved by using a profile in Maven:

 <project>
  [...]
  <profiles>
    <profile>
      <id>no-unit-tests</id>
      <build>
        <plugins>
          <plugin>
            <groupId>org.apache.maven.plugins</groupId>
            <artifactId>maven-surefire-plugin</artifactId>
            <configuration>
              <skipTests>true</skipTests>
            </configuration>
          </plugin>
        </plugins>
      </build>
    </profile>
  </profiles>
  [...]
</project>

So your first requirement:

1. If I run mvn install, I want all tests to compile, but I do not want to execute any.

can be achieved by using the following:

mvn -Pno-unit-test test

2. If I run mvn test, I want all tests to compile, but execute only unit tests.

This can simply achieved by using the plain call:

mvn test

cause the integration tests phase is not run (see the build life cycle).

3. If I run mvn integration-test, I want to compile and execute all tests.

This means running the default which includes running the test phase which will run the unit tests (maven-surefire-plugin) and furthermore running the integration test which are handled by the maven-failsafe-plugin. But you should be aware that if you like to call the integration tests you should using the following command:

mvn verify

instead, cause you missed the post-integration-test phase in your previous call.

Apart from the above you should follow the naming conventions for unit and integration tests where unit tests should be named like the following:

<includes>
 <include>**/*Test*.java</include>
 <include>**/*Test.java</include>
 <include>**/*TestCase.java</include>
</includes>

and integration tests should be named like the following:

<includes>
 <include>**/IT*.java</include>
 <include>**/*IT.java</include>
 <include>**/*ITCase.java</include>
</includes>

I hope you have configured the maven-failsafe-plugin like the following which is needed to bound the maven-failsafe-plugin to the correct life-cycle-phases:

<project>
  [...]
  <build>
    <plugins>
      <plugin>
        <groupId>org.apache.maven.plugins</groupId>
        <artifactId>maven-failsafe-plugin</artifactId>
        <version>2.15</version>
        <executions>
          <execution>
            <goals>
              <goal>integration-test</goal>
              <goal>verify</goal>
            </goals>
          </execution>
        </executions>
      </plugin>
    </plugins>
  </build>
  [...]
</project>

as you correctly did, but you should be aware that the include tags work on the source code (.java) and not on the compiled names (.class). I wouldn’t use the Category annotation, just simply using the naming conventions makes the pom simpler and shorter.