Part of the complication goes to two separate definitions for the word:
To be frequent (Miriam-Webster)
The most frequent, the most occurring wind direction in your example, even if weak, is east. So that would be the prevailing wind, as that's the definition we generally use in meteorology.
So don't think of it as
Having superior power or influence (Dictionary.com)
as indeed, that'd become a bit more subjective of a concept.
Sometimes you may hear about the average direction (which would be east-northeast in your example). So perhaps that's more what you're wishing for? Of course if you have an east wind for half the day and a west wind for half the day, the average wind direction would be either north or south. Or, if instead of treating direction and speed differently, you took the vector average... you'd have a mean windspeed of nothing!
Looks like this MathSE question fairly discusses some of the benefits and troubles with different methods of summarizing wind direction... which line up closely with the more general disparities of each of mean, median, and mode.
Alternatively, wind roses can be much more informative and enticing. They give both an indication of the directions (by the length of the bars in each direction) and the intensity (color coded). You can find things, such as: in Colorado Springs in May (as shown below), winds aren't from the southwest very often... but when they are, they tend to be fairly strong (whereas when winds are from about east, as is more common, they're typically quite light):
Looks like you can make some pretty snazzy/customizable recent wind rose plots of locations around the world at the Iowa Environmental Mesonet site.
But the take home is: in meteorology we typically define prevailing wind as simply the most common direction, with no consideration of the speed distribution.