The first force you mentioned is usually called the "strong force". It takes place between "baryons" (such as protons and neutrons) whether they are in a nucleus or not.
The second force is electrostatic, as you mentioned. It is part of the electromagnetic force. It is better to call it the electromagnetic force, because the electrostatics and magnetism are strongly related by relativity rules.
In an atom, there is a force that holds the protons & neutrons in the nucleus. Is this force called the nuclear force?
What about the force of attraction between the nucleus and electrons? Is is called the electrostatic force of attraction?
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thank you.