Brun's sieve bounds Working from Halberstam-Richert they state the following bounds
  \begin{align}
    S(\mathcal{A}; \mathfrak{P}, z) \leq XW(z)\left(1 + 2 \frac{\lambda^{2b + 1}e^{2\lambda}}{1 - \lambda^2 e^{2 + 2\lambda}}\operatorname{Exp}\left((2b + 3)\frac{c_1}{\lambda \log z}\right)\right) + O\left(z^{2b + c_2}\right)
  \end{align}
  and
  \begin{align}
    S(\mathcal{A}; \mathfrak{P}, z) \geq X W(z)\left(1 - 2 \frac{\lambda^{2b}e^{2\lambda}}{1 - \lambda^2 e^{2 + 2\lambda}}\operatorname{Exp}\left((2b + 2)\frac{c_1}{\lambda \log z}\right)\right) + O\left(z^{2b - 1 + c_2}\right)
  \end{align}
where $0 < \lambda e^{1 + \lambda} < 1$, $b \in \mathbb{N}$ and
  \begin{align}
    c_1 &= \frac{A_2}{2}\left(1 + A_1\left(\kappa + \frac{A_2}{\log 2}\right)\right)\\
    c_2 &= \frac{2.01}{e^{2\lambda/\kappa} - 1}
  \end{align}
and say that, if $1 < z < B_3$, where $B_3$ is some sufficiently large constant, these come from 
\begin{equation}
      S(\mathcal{A}; \mathfrak{P}, z) = XW(z) + \theta e^{A(2 \operatorname{li} z + 3)},
\end{equation}
where $|\theta| \leq 1$.
There are other constants and functions involved but, I suspect, they're not necessary to establishing this result. I just can't see how the bounds follow from the equality.
 A: Halberstam and Richert's Sieve Methods contains a sieve which consists of a pair of bounds on $S(\mathcal{A}; \mathfrak{P},z),$ their Theorem 2.1 on page 57. If we call the first bound in the OP (1) and the other (2), Halberstam says at the beginning of his proof of (1) and (2):

"We may assume that $z\geq B_3,$ where $B_3$ is sufficiently large, since otherwise our theorem follows from 3.14."

Line 3.14, with $0<|\theta|<1,$ is
$$S = S(\mathcal{A};\mathfrak{ P},z) =  XW(z) + \theta\cdot e^{A(2 li(z) + 3)}.$$
So for (1) the reader wants at least an argument that 
$$(*) \hspace{7mm} S = XW(z) + \theta\cdot e^{k_1 li( z) + k_2} < XW(z)(1+ k_3e^{(k_4/\log z)})  + O(z^{k_5})$$ 
for permissible $k_i,$ for $z$ not exceeding some finite number, $XW(z)> 1,$ and similarly for the second inequality (2). 
Halberstam's assertion for (1) depends  on the behavior of the second terms in (*) since 
for the first terms
$$XW(z) \leq XW(z)(1+ \text{anything positive}). $$ 
As for the second terms, one grows as $e^{li(z)}$ and the other as $z^k$ for some constant $k.$ Meanwhile $(1+ k_3e^{k_4/\log z})$ grows as $e^{1/\log z}.$ Eventually we expect that $e^{li( z)}$ will overtake both terms, reversing the sense of the inequality. Beyond that point we need a better S and 2.1 to preserve the original sense of (*).  
If for permissible constants the inequality (1) holds at all, we call the point where it fails $B_3$ and the authors' claim is apparently correct. As illustrated below, the $e^{k_4/\log z}$ term is initially very large for permissible constants, so we have good reason to expect that (1) will hold for small $z.$
Sample calculation.
Let the constants be as chosen in the first application of 2.1 (page 62) since they automatically  satisfy the imposed conditions $R, \Omega_1,\Omega_2(k)$ and are otherwise permissible.$^1$ 
These were chosen with (2) in mind so it's a poor but permissible example. 
$c_1=\frac{A_2}{2}(1+A_1(\kappa+\frac{A_2}{\log 2}))= 1(1+3(2+2/\log 2))\approx 16,$
$b=1, \lambda = 0.253, A_1= 3, A_o = \kappa = A_2 = 2, c_2 \approx 7, A = 2.$ 
For the first  terms in 3.14 and  (1) we have roughly
$$(**) \hspace{10mm}XW < XW(1+  0.98e^{\frac{309}{\log z}})$$
and the quantity in parentheses is very large for small values of $z.^{2}$ 
The quantity $.98e^{\frac{309}{\log z}}$ is so large that, even taking $\theta = 0.5,~\theta\cdot e^{4li(z)+6}$ does not catch up to it until roughly $z=48,$ ignoring if possible the factor $XW(z)$ which in applications is much larger than 1 (for $X= x > log^2 z,$ see pp. 32, 51).  
A similar argument applies to (2), bearing in mind signs of the remainder terms (4.5, page 57), but it's messier because the lower bound in (2) may be negative. 
The authors do not claim $z<B_3$ gives a useful sieve and I think this is why the claim is relegated to an unproved comment. A proof could be as complicated as 2.1 itself (6 pages + 4 pages of "technical preparation") and would boil down to something like the above, if it's correct.
$^1$ $\lambda =.253$ is used on page 63. 
$^2$ It's big but on page 66 the authors do the calculation for 2.1, (2) and get $\exp(\frac{8}{\lambda \log z}) =\exp( \frac{31.6}{\log z})$ so these are the sorts of numbers involved. Their estimate of $c_1=[1,2)$ seems low in light of chosen constants and 4.20 but it doesn't affect the result.   
Note: This is new material to me and I have taken the question as an opportunity to get better acquainted with 2.1. Corrections welcome, and my delete-threshold is a bit low.      
