In the US, a "mobile phone" is a cordless phone with a base.

As an American, I don't agree.

All the companies are branding as "mobile." I don't think they really even sell household corded phones anymore, we don't have to be specific about them.

If I wanted a corded phone, I think I'd have to use that specific modifier to get one.

Anyway, back to the point--say "mobile phone" to Americans and they know it's a cellular phone.

Seconded. "cellphone" and "mobile phone" are basically interchangeable to me. Though, honestly I'm more likely to just refer to it as a "phone" and use a qualifier to refer to the non-mobile variety (e.g. "house phone" or "land-line")