Name of this jar

Its a butter bell / butter keeper. My parents have one. You put water in the bottom, butter in the bell under the top/lid and that forms an airtight seal allowing you to keep the butter at room temperature.

Here's one for sale showing both parts clearly.

http://www.kitchenandcompany.com/Storage-And-Cleaning/Countertop-Organization/Butter-Dishes/_/Norpro-4-Butter-Keeper?tc=gfs13&utm_source=google&utm_medium=cpc&utm_content=Norpro%204%22%20Butter%20Bell%20Butter%20Keeper&gclid=CM2C1ZSRmroCFSpk7Aodt3kAgw

I've found a cookie jar buying guide on Walmart.com that has some interesting tidbits. For functionality alone, looks like a glass jar should do the job well. Something like this would have an advantageous shape to make cookies easier to reach, and the jar less prone to falling.

One feature you may want to look for specifically would be an hermetic sealing lid, which would help keep moisture in the jar and slow down the drying. Same kind of jar that is often used to store flour, sugar, coffee, beans, etc.

Is light sensitivity a factor?

I don't think it would be, in most cases, unless you kept your cookies in an area that is heavily lit by the sun.