Enlightened self-interest seems like what you are looking for.
Enlightened self-interest is a philosophy in ethics which states that persons who act to further the interests of others (or the interests of the group or groups to which they belong), ultimately serve their own self-interest.
It has often been simply expressed by the belief that an individual, group, or even a commercial entity will "do well by doing good".
A further explanation from pscyhologytoday.com:
Enlightened self-interest refers to the understanding and trust that what a person does to enhance another’s quality of life enhances one’s own quality of life to a similar degree. More simply put, it is the idea that “what goes around comes around.”
While most of our actions are motivated by a desire to fulfill inner needs and personal desires, acts of enlightened self-interest serve the well-being of others as well. Enlightened self-interest means that everybody wins.
The recognition that being committed to another’s well being is personally fulfilling is the basis of a cycle of mutual generosity that creates an ongoing, self-reinforcing loop that deepens and becomes more enriching over time.
There is a similar question but not asking the same thing:
What's the antonym for Schadenfreude?
In your question, you emphasize that you are satisfied by actually doing something and helping people instead of just being happy when others are happy. In the latter case, they might be happy for another reason. So, this is how enlightened self-interest differs from mudita.