No interference is expected between clients of different types, as Html or Java.
The difference in the set of subscribed items should be the only significant one from the Server perspective.
Anyway, no interference is expected between the two clients as well;
the Server treats all items independently from one another.
In your tests, what changes is the sequence of "subscribe" and possibly "unsubscribe" invocations to the Data Adapter for the various items;
you should ensure that the Data Adapter also treats all items independently from one another and that it can handle possible sequences of "subscribe"-"unsubscribe"-"subscribe" invocations for the same item.

The whole processing of items and updates can be logged if you set the "LightstreamerLogger.subscriptions" and "LightstreamerLogger.pump" loggers at DEBUG level.
Setting up a minimal test case is useful but not necessary in order to analyze this log.