Lease dispute and seek suggestion in Seattle, WA (Washington)

[Ellen Li]

My lease agreement with tenants has expired on May 21st and in the lease there is a extended period for month on month lease after the lease expired, which is assumed to be from May 22nd, 2017. Today my tenant sent me a notice that they think the month on month lease starts on the first of the month so they only need to give 20 day notice which is before June 10th to inform me whether they will stay in the house after June 30th. In the lease it has clearly stated the lease starting date was August 22nd 2016. The rent is due and paid in the 1st of every month but it is not the start of the lease month. The tenant is trying to give me the notice as late as possible and also give away the obligation to pay for the full month rent because of their late notice. The tenant has proper job and income and it's just that they are selfish and don't care about my schedule. I am planning to go for small claim if they don't pay me full month rent and wonder if you have any better suggestion? Also, does tenant's credit get harmed by being filed with no pay or late rent payment? Appreciate any advice or recommendation of landlord lawyer in Seattle.