The reason to have an immediate past president serve in the governing structure of a nonprofit corporation is usually to assure that the experience, and perhaps the wisdom, of the most recent former president is available for consultation and participation in the governance of the organization.  That purpose seems to be eliminated if that experience is lost simply because the new president serves a second one-year term.  If the bylaws intended to limit the service to a single year, they easily could have said that the immediate past president serves on the executive committee for a year after completion of his or her service as president.

You should also look back at the history of your organization.  Did your immediate past president serve more than a year on the executive committee when you served as president of the organization?  What happened with the prior transitions?  Is this a unique interpretation in your history?

Although I don’t know the history of your organization and haven’t seen the full bylaws, it sounds to me as though you should continue as immediate past president on the executive committee.  (I see from your website that you are listed as a current officer and perhaps, depending on the specific language of the bylaws, you have a seat on the committee by virtue of your current position.)  If the other officers don’t think you should serve, you could seek support from the other directors.  They may be able to help get this straightened out.