fa.functional analysis - Saito-Wright definition of Rickart C*-algebras

A C*-algebra is Rickart if for each $xin A$ there is a projection $pin A$ so that
$R(x)=pA$.
Here the right-annihilator $R(S)$ of $Ssubset A$ is defined
as $$R(S)={ain Amid xa=0, forall xin S}$$ and $R(x)equiv R({x})$.

In:

Kazuyuki Saito and J. D. Maitland Wright. $C^∗$-algebras which are Grothendieck
spaces. Rend. Circ. Mat. Palermo (2), 52(1):141–144, 2003.

an alternative definition is studied:
define a C*-algebra to be Rickart if each maximal Abelian *-subalgebra of $A$ is Rickart (or, equivalently, monotone $sigma$-complete).
Equivalently, one may require that every Abelian *-subalgebra is contained in an
Abelian Rickart C*-algebra.

This definition is more general and seems to be sufficient for many applications.

Is this definition in fact equivalent to the original one?

This question recently came up in our investigations in the foundations of quantum theory:

Bohrification of operator algebras and quantum logic