Is there a term for the personal wealth that is not used to generate income?
I do not believe that your suggested definitions will hold up. Since this is a forum for questions about economics, I do not see the point of proposing a new definition here. Who is going to see it?

What you call “income generating wealth” is captured by “wealth” in its standard usages. However, what you exclude (such as bank deposits) will be included in “wealth” or “claims on wealth.” From a portfolio perspective, it makes no sense to exclude bank deposits. Furthermore, “income” is well defined, and there are assets that generate no income for the holder, and yet are wealth (for example, a personal residence). Rephrasing to include future income does not help, since the sale of a primary residence is not considered income.

Everything else would be classified as “consumer goods.” If you want to only include things that “make you look like a rich person,” you would end up with the fuzzy category of “positional goods.”

Why not include consumer goods in wealth? One of the regularities of consumer goods is that they have almost no resale value. So there’s no good way to measure its value.

The lines are blurry. For a collector that buys and sells particular goods, those goods are actually assets that are expected to make a profit. However, no statistician could hope to measure the value of these exceptions.