I have some ordinal and nominal examples for SPSS and am not sure whether I have done it right.
Ordinal is where order is important e.g. Height: 1 is 150cm-159cm, 2 is 160cm-169cm, 3 is 170cm-179cm, 4 is 180cm-189cm
Another ordinal example: 1 is high, 2 is medium and 3 is low
Nominal is where order doesn't matter e.g. 1 is vegetables, 2 is fruit, 3 is dairy, 4 is confectionery
Your examples are fine except that they leave out the third case - scale variables. Your first example might be treated as scale. You might find it useful to read Ruben van den Berg's recent article on measurement levels at https://www.spss-tutorials.com/ (Look for the measurement level tutorial).
However, in most cases Statistics does not actively use measurement levels except for categorical vs continuous/scale in a few cases. TREES is an exception. It is up to the analyst to use the appropriate techniques. Statistics will guess the measurement level when data are read, but these are only guesses.