Is it truthy or falsy?
MATL, 20 bytes
U|sG36ZA13\[BID1]m+g

Input is a string enclosed in single quotes. Output is 1 for truthy or 0 for falsy.
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How it works
U|s      % Take input implicitly. Interpret as number, absolute value, sum
G36ZA    % Push input again. Convert from base-36
13\      % Modulo 13
[BID1]   % Push array [6 3 8 1]
m        % True if member: gives 1 iff the result from modulo 13 is in the array
+g       % Add, convert to logical. Display implicitly

This performs two tests on the input:

Try to interpret the input as a number, and detect if it is nonzero. The function used to interpret a string as a number outputs an empty array if it's not possible; and the sum of the entries of an empty array is 0. So it suffices to try the conversion, take the absolute value, and sum. This gives a positive value if the input contains a nonzero number, and 0 otherwise.
Assuming the input string doesn't represent a number, we need to classify it into one of the two given sets. To do this, the input is interpreted as the digits of a number expressed in base-36, using alphabet '01...9ab...z'. The base conversion function is case-insensitive and ignores digits not present in the alphabet (in our case, '-'). It turns out that the modulo 13 of the resulting number is 1, 3, 6 or 8 for the truthy strings, and doesn't give any of those values for the falsy strings. So it can be used as a signature. We thus perform modulo 13 and see if the result is any of those four values. This gives 1 if it is, or 0 otherwise.

The final result should be truthy if any of the two conditions is met, and falsy otherwise. So we add the two numbers resulting from 1 and 2 above and convert to logical, which gives 1 or 0 as standardized truthy/falsy values.