How to create test cases for the registration form?

Question

Software testing

For example, there is a registration form with many fields. If you do not fill in (or fill in invalid data) all the fields and click [Submit], then an error will be displayed under each field, if you fill in all fields with valid data except one, then only one field will display an error.
How is it correct?
1) Check that all errors were highlighted immediately by filling in all fields with invalid data or leaving them empty? (i.e. there will be a maximum of 2 test cases)
2) Check that an error has appeared under field n after filling all fields except field n with valid data. Repeat n times for each field. (Many test cases)