Since Lowengard said much of what I was thinking, I'll try to help with your last question/concern:

Thank you so much for your response, I really appreciate it! I'm thinking along the same lines as you. I just need to figure out how I approach the company, since they think I'm working on the design. Since I offered the whole package, I figured that was the plan. But after the company said to go ahead, then I found out about the other designer's intentions of making the updates.

To be honest, I'd give them a call, or better, schedule a meeting with the people you met with initially. If they want to include they're web designer, fine, but I wouldn't request that he/she be present. At your meeting, you mention your concerns and the stipulations by which your offer was based. Then simply let them know you appreciate the opportunity to assist them but that you cannot offer the services for free. You made it clear from the beginning that the offer included site maintenance, and without that, it's not valid. Good luck and let us know how it works out.