Under that definition, $1/year/person would still "qualify" as UBI, and I don't think there are many interesting conversations to be had around that.
This is why I'm glad in practice politicians, economists, etc., tend to study these things in order figure out how to structure social services in a way that actually helps people and not to make "Reductio ad absurdum" style arguments.
It sounds like you're conflating a few things. UBI is as it says, an income that is given universally. UBI doesn't imply a certain amount of income. Other phrases may be used for what you seem to be describing: full basic income, guaranteed minimum income (which implies means testing).