In some places, I find the following definition of stable distribution:
A distribution is said to be stable if a linear combination of two independent random variables with this distribution has the same distribution, up to location and scale parameters.
And in some places,
Let X1 and X2 be independent copies of a random variable X. Then X is said to be stable if for any constants a>0 and b>0 the random variable aX1 + bX2 has the same distribution as cX + d for some constants c>0 and d.
In the first definition, we don't have any constraints on $a$, $b$ and $c$; they need not to be strictly positive. So, are these two definitions equivalent?