These words are related by derivation. In each case, one of the parts of speech was first. Then others were created from one or more of its senses, but can then take on a semantic life of their own.
(In English it is very subtle because the surface forms are identical or, in your life N / live V / live ADJ example, very similar and the derived by a process that is no longer productive.)
Because of constant semantic shift, the senses are hardly ever 1:1 even on day one, so it is dangerous to define concepts as if they are logical and orderly, not fuzzy and tangential.
live ADJ is in the same semantic field as television and concert, but in some other languages live in the sense of television is literally direct ether, and literally alive or living is only used for in-person performance as opposed to something remove.
(In English it's a bit hard to make this distinction, so for example in a café full of screens showing the Eurovision final in real time, it's live but not live.)
This hints at us that this sense of live is not very tightly bound to live V, and, in fact, upon thought it's true, show and concert have nothing special to do with live V or life N, nor with lifer or leave, which are also derived from the same root.
Another proof of this is that with enough derivations, we can get back to the same part of speech, yet they are not equivalent. lively is not the same as alive, enliven is not the same as live, liveliness is not the same as life.
So dictionaries like Wiktionary take the safe route by listing a flat representation. (A lemma can have multiple senses, a sense can only have one lemma.)