I am wondering if I should explicitly refer to processor in software method claims, like "...comprising executing on a processor the steps of..." or it is enough to say "computer-implemented method" to cover processor, storage, memory, input/output etc.
Assuming this is in the U.S., start with only "a computer implemented method" because it's broader than reciting that a processor performs the steps. Better yet, simply recite "a method" (which is broadest) and then wait until examination to make any further limitations. The Examiner may or may not require further limitations.
Firstly it depends on your definition defined in specification and limitations associated with it. Based on State of the Art and judicial claim construction i believe "computer-implemented method" have broader interpretation and careful drafting of dependent claims and specification will cover all your objects 'processor, storage, memory, input/output etc.' alternatively You can use preamble 'A Method'.
In Atser Research Technologies, Inc. v. Raba-Kistner Consultants Inc., et al., 2009 U.S. Dist. LEXIS 25294 (W.D. Tex., Feb. 27, 2009). It was stated that "computer-implemented method" will cover steps performed using a computer, which include processor and other peripherals.
On the other hand 'executing on processor steps' will cover all steps performed in processor only. You can improve processor definition or phrase as such by including all component used in steps.