Worrying that a LL doesn’t know this stuff. And I speak as a LL.
Surely you’d check-it out BEFORE giving notice, using at worst a forum for Landlords, rather than general MN site where often it’s the blind leading the blind.
As a LL, there are the legal requirements which must always be met. And then additionally, there is being sensible and treating people decently, which is always a very good idea too. You want and benefit from the goodwill of tenants and therefore need to show them some of that too.
In this scenario, personally, I’d have thanked the tenant for being willing and able to vacate before the date I’d given, and in that scenario, I’d probably say I’d refund the last week or so (beyond the usual rent cut off) if they vacated when they said they would.
OP doesn’t seem to realise how tenants often don’t vacate when they say they will, and actually can choose not to vacate by the end of notice period and that then legal processes have to occur to evict. It is a tenant’s legal right to avail themselves of this process and remain. It can make things very difficult for a LL. So, given you’re already serving notice which is pretty annoying, although perfectly legal, and given the tenant is being helpful and has offered a date sooner than the last, I’d be looking to acknowledge this and encourage it to actually happen by offering to return the last week of rent as a token of appreciation. Such a gesture can lead to goodwill, a cleaner flat etc etc.
I’d also say, that if a tenant has been there for a decent period of time, when it comes to them moving out, unless there was genuine serious damage, I would not be looking to deduct from the deposit for minor things. Some people report LLs wanting to withhold parts of deposit for dust on top of wardrobes, dirty windows etc. Perhaos with a short term let or someone who hasn’t paid their rent etc, you’d want to insist on the letter of the contract regarding deposit returns, but if someone has been there for 3 or 4 years and paid tens of thousands in rent, I’d be returning their full deposit sharpish and disregarding most things and just considering them fair wear and tear, even if they might be a bit more than that.