Example: The word stellar contains the word star. The word sanguine contains the word sun.
A word embedded in the manner you mention (as, for example, star within stellar, or sun within sanguine) can be said to be a subsequence:
In mathematics, a subsequence is a sequence that can be derived from another sequence by deleting some elements without changing the order of the remaining elements. For example, the sequence <A,B,D> is a subsequence of <A,B,C,D,E,F>.
Since a word is a sequence of letters, one might by analogy with subsequence speak of a subword, except that in wiktionary subword is defined as "(mathematics) substring", and substring implies contiguity of symbols, which subsequence does not.
Unfortunately, as anyone can see, star is not "contained in" stellar, nor sun in sanguine. So there may well not be a word for such a non-existent effect. If you're talking about Etymology, maybe there is an effect, at least with stellar, though it's certainly not any variety of containment.
The English word stellar is borrowed directly from the Latin word stellaris, an adjective meaning 'pertaining to stars' that is formed from the Latin root stella -- which means 'star' -- plus the adjective-forming suffixes -ar/al-is. But that's a Latin word containing a Latin root, formed by Latin rules around the year 0. Then, 1500 years later, the word stellar (but not the word stella) was borrowed into English.
So it's a little more complicated than "containing" might suggest. Words are not things, after all, like leaves or fingernails. Words are transitory patterns of behavior that are alive as habits in the body memory of speakers. Words don't have insides or outsides, so the idea of one word "containing" another is as likely as your headache containing your upset stomach.
Words do have histories, but the histories have to be correct in order to be useful, and worthy of special terminology. For instance, there is no mention, reference, history, or relation between the words sanguine and sun.
Sun is from an old Germanic word that's been in the 'sun' business for millenia; it even has its own day of the week in Germanic languages.
Sanguine, on the other hand, is borrowed from Latin; it originally meant 'bloody', and came from the Latin word sanguis, which means 'blood'. Then it changed to mean a blood-red color, and particularly a reddish facial complexion, which in English people -- who were all white at this time -- was a sign of health in their damp cold climate. From there it came to mean 'enthusiastic', 'cheerful', and -- most recently and most commonly now -- 'optimistic'.
So, a cheerful, enthusiastic, optimistic person with a reddish facial complexion might well be said to have a sunny disposition, too. It's compatible, as many metaphors are. But no word "contains" any other word here.
Executive Summary: Etymology is interesting, but get it right before you look for terminology.