Ring LWE: How can the secret be chosen from the "uniform distribution"?
You're right that essentially all schemes require that the secrets are small in a concrete sense.
While I'm not terribly familiar with the standard, I believe the tables in question were generated by the LWE Estimator of Albrecht et al. This page of its documentation states various secret distributions they support. I expect "uniform" corresponds to "bounded uniform", but I'm unsure of which particular bound was used to generate that table.