For $x$ a self adjoint element of a $C^*$ algebra it is equivalent:

1. $x$ has non negative spectrum
2. $x$ has a square root $x=y^*y$
3. $x$ is a finite sum of squares $x=\sum {a_i}^*a_i$

in this case $x$ is indeed called positive.

For $x$ a self adjoint element of a $C^*$ algebra it is equivalent:

1. $x$ has non negative spectrum
2. $x$ has a self adjoint square root $x=y^2$
3. $x$ is a finite sum of squares $x=\sum {a_i}^*a_i$

in this case $x$ is indeed called positive.