If they are admitting that they received the termination notice, but are arguing that it was in improper form, you do have a decent claim that you complied with the essential requirements of the lease. If they deny receiving the post-it note, you would have a harder time making that claim - as you would have to somehow convince a court that they did receive it despite their denials. (The "if it fell off, they could have compared it to all of the signatures in their files" argument is probably not going to fly in court - that's a lot of work to figure out who submitted an improper notice, assuming it was even processed and not simply swept up as trash.)
It is not clear upon what basis they are tacking on two months rent. First, if you only have to give one month's notice, it seems per se unreasonable to attach a two month penalty to improper notice to quit. Second, if they had a tenant in place during that time, they're effectively trying to double-dip.

