The key to what it's all about is in that phrase "local support adjustment".
As you, and everyone else, have found, you were entitled to a 25% discount in your Council Tax Bill (as the sole adult occupier) up until your daughter turned 18. You've been given that and it's removal (which never occurred prior to August anyway) is NOT the reason for your revised bill.
You obviously also claim for Council Tax Support, which replaced Council Tax Benefit and is administered locally (which is why there's a reference to 'local support' on your bill). The level of support you receive depends upon your family circumstances, such as your household income and the number of dependants that you have. What the council have said is that, since your daughter is over school-leaving age and no longer in full-time education she ceases to be regarded as your dependant.
So you've NOT lost your single-occupier Council Tax reduction (up until August). You HAVE lost some (or all) of your Council Tax Support, for what appears to be a perfectly valid reason.
To put some figures on that:
Someone might have a 'full' Council Tax bill for their property of £1000 per year but, because they're the only adult in the house, that's brought down by 25% to £750 per year. They then apply for Council Tax Reduction and the council reduce the bill by a further two thirds, bringing it down to just £250 per year. However that discount was based upon the occupier having a dependant child to care for. Once that child ceases to become 'dependent', because they're no longer in full-time education, the council reduces the support provided, possibly by halving it, and bring the bill back up to £500 per year.
Any clearer now?