I might not have understood correctly, but this sounds more like the price of the curve trade rather than the carry. On this note, yes, a 5s30s steepener will be cheaper than a 30y payer outright.
"the rate you are receiving on your short-end long, net the rate you are paying to short the long-end."
I would think the carry in the report you read has a different definition. The carry for a (assumed) 6m horizon is the certain payment that is earned during that period.
Using your 30y paid position as an example, and using the notation R(term, tenor), the 6m carry on it is $$ Rate(0, 30y) - Rate(6m, 29.5y) $$
If the curve is upward sloping just prior to the 30y point, then your carry on the 30y payer position is negative. The report you have implies that a 5y receiver position has positive carry, so that makes the carry of the steepener position less negative than the outright. I don't have actual numbers with me, but you can plug the values into the formula above to calculate the actual carry values to validate the report's claim.
Note that the formula is for a payer position, for a receiver position it's the forward minus spot.
Related to carry is the roll down, and sometimes they're bundled together commonly as carry + rolldown, though some sources may just refer to it as carry so you might have to look out for any footnote or definitions that are set out by the author.