I think you've hit on one of the reasons why factions are often so difficult for kids. They are so multi-dimensional. You are certainly right that kids don't recognize that a fraction can be interpreted as a division problem. A whole or a set has somehow been divided into equal parts and those parts are thirds. They should have a solid sense that thirds means three parts.
Parts means something has been divided.
It could be an any size something that is being divided, but there are always three equal parts. There are many, many activities to help build this type of thinking. It will help the children remember. I put it on the wall in the Title 1 CAI area. I never got any feedback and have no idea if it ever helped.