No you are getting muddled between two schemes.
If two or more adults live in a property, the full Council Tax charge is payable on that property. If one adult lives in a property, the Council Tax charge is reduced by 25% (there are various rules about who counts as a adult - full-time students don't for example - but that's the gist of it). So the full charge will be chargeable on the PROPERTY because there are two adults living there - this is nothing to do with household income or the rent paid to anyone.
If the income of the household is less than certain thresholds, there may be a discount off the assessed CT charge. This is a means-tested benefit and is where the bit about having savings comes into it.