I assume that, since Japanese has topic-orientated sentences, rather than subject-orientated, this construction is closest to their way of thinking.
"Watashi wa, (Concerning me) .... I am happy today." Some other languages use this construction too:
Topic, <topic particle>, information about topic.
This not how we think or talk in English. We do not need to state the topic first, and then elaborate on it. Giving them an example of the right way to speak like this is simply avoiding the issue. The right approach is not to use it at all, since it does not generate proper colloquial English sentences.
Instead of "In the case of me, I usually stay at home if it's raining." explain to them that in English, we say simply, "I usually stay at home if it's raining". Encourage a conversation. When someone begins with "Concerning rain, ..." or "In the case of rain, I ..." stop them; emphasise that they do not need to restate the topic. Encourage variations:
"What I like to do when it's raining is to ...."
"The last time it rained heavily, I ... "