The statistical programming language R uses the term "ordered factor," so factor isn't completely safe, though I can't find an example of an ordinal variable being called "factor" without the adjective.
I think ordinal variables are fairly often considered either qualitative or quantitative: they have the quantitative feel from the ordering, but lack mathematical operations. See the end of this post for a few examples.
But "nominal" seems relatively safe as meaning only unordered. (I found one contradiction to this, the Bonus Note #2 at https://www.mymarketresearchmethods.com/types-of-data-nominal-ordinal-interval-ratio/, but that's immediately contradicted by having "ordinal" in the next section?)
See also https://en.wikipedia.org/wiki/Level_of_measurement, especially the "Debate" section that lists a few other proposals. (Chrisman's proposal is nice for including "cyclic" features that are sometimes important in ML but don't fit into most standard libraries without some [unfaithful] encoding.)
A few links to show that the lines get blurred: