饭盒 is a general term referring to any kind of box in which food of meal is carried.

There are many kinds of materials that can make 饭盒.

In restaurants, since the 饭盒 is for customers to take away, they must be very cheap.

So, the materials of those 饭盒 are styrofoam and paper. Paper is much more environment-friendly and now getting more and more pupolar.

In my opinion, I think 饭盒 is the correct word for the Styrofoam box IN RESTAURANTS.

Generally speaking, there are two ways to take your left food out in Chinese type of restaurants: 饭盒 and 塑料袋 su4 liao4 dai4 (plastic bag). Sometimes people would like to only choose one of them, sometime people maybe will use both of them: 饭盒 for food and 塑料袋 for carrying 饭盒 which contains food. In some foreign type of resturants or fast food shops, such as KFC, they probably like to use paper bag(this is the doggy bag what you called, isn't it?) And currently we also have some Chinese fast food shops, such as 一品三笑 yi pin san xiao, besides 塑料袋, these shops like to use a kind of transparent plastic box(I don't know the Chinese name, sorry). Sometimes they like to call it as 饭盒, sometimes they maybe will call its real name.

If you mean lunch box, I think is more like the box we can see in the supermarket for taking food which you we prepared at home to go to school or go to work. We also call it 饭盒，but in real Chinese restaurants and fast food shops, we never use this kind of 饭盒to carry food. And the Styrofoam box can be considered as the only 饭盒 in these places.

Plus: I think maybe you have known, but I still want to remind you that don't call 饭盒 as 盒饭。 盒饭 means Chinese take-away or packed meal, 饭盒 means the box to put food in.

May these are helpful.

饭盒 is very general - it refers to any box that you use to put food in, including the examples you mentioned and including a lunch box that you would take to school/work.  If you want to refer specifically to what they use in a restaurant where you can take the food home, it's 打包盒 dǎbāohé (to-go box) or 打包带 dǎbāodài (to-go bag/doggy bag)