Census Designated Places

The Census Bureau considers the area under discussion to be a census designated place (CDP). They explain what a CDP is here, including this explanation for how a CDP should be named.

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7. The CDP name should be one that is recognized and used in daily 
communication by the residents of the community. Because unincorporated 
communities generally lack legally defined boundaries, a commonly used 
community name and the geographic extent of its use by local residents 
is often the best identifier of the extent of a place, the assumption 
being that if residents associate with a particular name and use it to 
identify the place in which they live, then the CDP's boundaries can be 
mapped based on the use of the name. There should be features in the 
landscape that use the name, such that a non-resident would have a 
general sense of the location or extent of the community; for example, 
signs indicating when one is entering the community; highway exit signs 
that use the name; or businesses, schools, or other buildings that make 
use of the name. It should not be a name developed solely for planning 
or other purposes (including simply to obtain data from the Census 
Bureau) that is not in regular daily use by the local residents and 
business establishments.

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I figure "East Whittier" meets their criteria for a proper CDP name, and "East La Mirada" does not.