I might not have understood correctly, but this sounds more like the price of the curve trade rather than the carry.

> "the rate you are receiving on your short-end long, net the rate you
> are paying to short the long-end."

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.*