For the name (word), I can write: “cat”, ‘cat’, or cat
For the referent, I can write: cat
For the meaning, how should I write it?
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The name is “cat” or ‘cat’
The meaning is “a small domestic animal with soft fur kept as a pet or for catching mice.” or ‘a small domestic animal with soft fur kept as a pet or for catching mice.’
It is a matter of stylistic choice whether you use single or double quotation marks. Just as it is a stylistic choice whether you choose serif or sanserif font.