you are correct about it being the landlords responsibility (as per the law) but they very rarely (if ever) get fined or even checked. In practice, the responsibility falls with the tenant as they are the one requesting a service from immigration and therefore must ensure the TM30 is up to date to facilitate the processing of their request. The tenant is also the one who usually pays the fine as they are the ones having to face immigration.
retain your spent boarding pass in to Thailand as proof of incoming flight and hand that over to the Immigration office with your passport. Then have your flight out booking confirmation printed to hand over as well. The 60 day visa exempt is a very simple, no fuss way to enter the country. You will have no issues my friend and enjoy Thailand.