How can I easily explain to a layperson the difference between types of "contains"
I would say something like this:

All matter -- that is, all substances -- are made of atoms.  The properties of a given substance are determined by (a) the type and (b) the arrangement of the atoms in that material.  

Chemists refer to the "types" of atoms as "elements".  For example, vegetable oil and table sugar (sucrose) are both composed of three types of atoms only: carbon, hydrogen, and oxygen.  

The arrangement of the atoms in the two materials differs drastically.  (You could refer to structure diagrams at this point if your friend is interested.)  First, oil has much less oxygen and slightly more hydrogen per carbon atom.  Second, the way the atoms are connected -- their arrangement -- is very different.

Chemists sometimes give a name to a group of similar arrangements of atoms.  For example, "sugars" are compounds that (a) have carbon, hydrogen, and oxygen atoms in ratios that are approximately $\ce{CH2O}$, and (b) often form ring structures, as seen in table sugar, glucose, and corn starch.  But the key words are "approximately" and "often" because some molecules that chemists call sugars don't satisfy these rules.  Biochemists interested in metabolism sometimes refer to molecules that can "easily" be converted into "true" sugars as sugars.  

Naming in chemistry is complicated by the fact that chemists use words for different purposes: to indicate metabolic similarity, to indicate structural similarity, or to indicate similarity in reactivity.  Those different purposes sometimes lead situation where chemist B says that what chemist A calls a "sugar"  isn't really a sugar.  But that's just an argument about humans' imperfect way of naming things, not about chemistry.  There is a lot of wiggle room in the words "approxiately", "often", "similar", and "easily".  So if we focus on metabolism only, the glycerol part of the vegetable oil arrangement (i.e. structure) is a bit like a sugar.  But it is dissimilar from sugars in terms of both structure and reactivity.

Again, what ultimately matters is (a) the type and (b) the arrangement of the atoms in a substance.  Those two things are together necessary and sufficient for a complete description of all substances.  Whether to call particular arrangements of particular atom types "sugars" or not is something people can argue about.