The Dual-Stack Test is meant to test whether your client is choosing IPv6 over IPv4 when making a connection to the server since it is known on the Internet with both IPv6 and IPv4 addresses. The proper behavior of your client, assuming that the IPv6-only test works for you, is that the Dual-Stack Test would have an identical result to the IPv6-only test and confirming that you are preferring IPv6 over IPv4 when connecting to a dual-stack destination. If the result is a page with a red box stating that you are using IPv4, then your browser and/or IP stack are preferring IPv4 over IPv6, which is undesired/broken behavior.
One type of test that seems to be worth mentioning in this thread is stress/performance/load tests which could be simply put as finding out the limits beyond which a certain piece of software breaks. Note that in terms of tooling it is essential to precisely determine the scope of what one proposes to stress tests from a system perspective. For instance in the case of a "web application" stress testing can include in its scope the web server application itself and so the tooling could intervene on that end. Here is a nice post about http load testing