Is it illegal to use a legal name? Lawful v legal and courts of equity.

Note the following are not official definitions but my interpretation of them and attempt to provide a more accessible meaning to the layman. Do not blindly accept them but rather consider and weigh them:

Lawful - in layman's terms that which is right and proper, pure justice.
Legal - in accordance with established rules, and therefore not of necessity, right and proper, in that the rules may be written to permit that which is not right and proper.

Thus that which is legal may not be lawful.

An example:
Lawful behaviour - to drive in a manner considerate of others, that is to drive safely and courteously
Legal behaviour - to drive in accordance with the rules, as to specific speed, signalling to change lanes, giving way, etc.

The speed limit is not lawful, it is legal. The lawful speed is that which is duly safe and considerate of others and may be either faster or slower than the legal speed limit which is an arbitrary rule.