If you scroll down the M-W page with the definition of livid that you provide, you will find this usage note that explains quite well the issue:
Livid has a colorful history. The Latin adjective lividus means "dull, grayish, or leaden blue." From this came the French livide, which English borrowed as livid. The word can describe flesh discolored by a bruise or an appearance deficient in color. Eventually, it came to be used for the complexion of a person pale with anger (i.e., "a person livid with rage"). From this meaning came two new senses: "reddish," as one is as likely to become red with anger as pale; the other was simply "angry" or "furious."
So this entry clarifies, that you cannot really describe a car as livid. This is an adjective that describes the colour of skin, the colour of bruise marks on the flesh and also the colour of skin provoked by anger, rage or fear.
Cambridge defines it as:
(esp. of marks on the skin) of a purple or dark blue color, usually caused by an injury:
- There was a livid bruise on her upper arm where she had fallen.
So if you hear a simple sentence out of context:
She was livid.
you will know that this either involves some physical injury that leaves marks or a strong emotion provoking that lividity either literally or figuratively. However, normally, this word is surrounded by context which clearly will explain its meaning. You will very often encounter expressions such as livid with fear/horror (pale), livid with fury etc. Here is more information from (Etymonline) which also shows that:
Somehow livid has come to be associated with "pale, colorless." The sense of "furiously angry" (1912) is from the notion of being livid with rage. Perhaps this is the key to the meaning shift. Rage makes some dark-red-faced; purple with rage is not uncommon in old novels:
'My money! ye pirate! or I'll strangle you.' And he advanced upon him purple with rage, and shot out his long threatening arm, and brown fingers working in the air.
(Hard Cash: A Matter-of-fact Romance, Charles Reade, 1869)
while it makes others go pale, also a figure in old novels:
At this juncture, the door opened, and, pale with rage, her eyes flashing fire, Lady Audley stood before them.
(Marriage. A Novel, Susan Ferrier, 1825)
Here is an interesting explanation of the word in the context of Scrabble: Win Every Game.
Lastly, note that some dictionaries (like Collins, for example) label the usage of livid to mean furious, extremely angry as informal, which is probably why it is the most common.