In a nutshell
There are two issues that justify the statement of your reference:
The free or bound character of a variable depends on how much context
you are considering, and whether it contains a binding occurrence of
the variable
A variable may be re-bound within the scope of an existing binding, so
that removing that binding does not preclude that some occurrences may
still be bound.
More detailed explanation
There are really several issues, some of which do not seem to be
emphasized by Gilles'answer, which is very good and need not be
duplicated here for the issues it addresses (as I understand it).
Being free or bound is a property of variable occurrences, not of
variables themselves. Note that the occurrence that follows the
$\lambda$ (in blue in Gilles'examples) is a binding occurrence, and is
neither free nor bound.
The other issue is that, for a given occurrence, being free or bound
is dependent on the part of the context you are considering.
If you examine the example:
$$\color{purple}{(}\lambda \color{blue}{x}. \color{red}{(\& (f_1 \color{green}{x})
(f_2 \color{green}{x}))}\color{purple}{)}A$$
the $\color{blue}{\text{blue}}$ occurrence of $x$ is a binding occurrence. The $\color{green}{\text{green}}$
occurrences of $x$ are bound by this binding occurrence whenever you
consider a context that includes the binding occurrence, such as the
whole $\lambda$-expression or just the part between the
$\color{purple}{\text{purple}}$ parentheses.
But if you consider a smaller context, abstracting away the part
containing the binding occurrence, as for example the sub-expression
in $\color{red}{\text{red}}$, the $\color{green}{\text{green}}$ (occurrences of the)
variable $x$ are free in this subexpression.
So being free or bound is dependent on the occurrence considered and
on the context considered.
But it may be said that variable $x$ is free in some (sub-)expression
when it constitutes a context where $x$ is never bound. Note that a
variable $x$ can be free for some occurrences and bound for others in
the same $\lambda$-expression,
depending on the scope of binding occurrences.
The confusion you are confronted with comes from the fact that when
you do the $\beta$-reduction, applying the function between
$\color{purple}{\text{purple}}$ parentheses to the argument $A$, you
replace the whole redex by just the part in $\color{red}{\text{red}}$
where the variable $x$ is indeed free, now that the binding occurrence
is removed, prior to making the substitution of $x$ for the arguments.
But the situation could be more complex. There could be several
binding occurrences of $x$ as in:
$$\color{purple}{(}\lambda \color{blue}{x}. \color{red}{(\& (f_1 \color{green}{x})
(\lambda \color{magenta}{x}.f_2
\color{green}{x}))}\color{purple}{)}A$$
Now both $\color{green}{\text{green}}$ $x$ are bound, but the first is
bound by the $\color{blue}{\text{blue}}$ binding occurrence, while the
second is bound by the $\color{magenta}{\text{magenta}}$ binding
occurrence.
When you do the $\beta$-reduction as before, you remove as before the
$\color{blue}{\text{blue}}$ binding occurrence of $x$ and substitute
the now free occurrences of $x$ for the argument. But only the first
green occurrence is free, while the second is still bound by the
$\color{magenta}{\text{magenta}}$ binding occurrence.
That is why your reference insisted that the variable occurrences were
(now) free for being substituted by the arguments. There could also be
non free occurrences, as in the example I just gave, despite the fact
that a binding occurrence of the variable had been removed.