You are correct. The article is flatly wrong when it suggests that people would lose the UBI as they start making money elsewhere. It is not global unemployment insurance.
There are certainly UBI proposals in which at some point UBI does scale back. I don't see a good reason why it scaling back would disqualify it from being a UBI, and I certainly see little likeness to unemployment insurance.
I guess the justification for not scaling it back would be that it makes it more palatable for the people paying for it. If someone received $10,000 in UBI and payed $10,000 in tax it would be neutral but their attitude to other recipients might be different.