TX Antenna dBi relevant for transmit and receive rate?
Simple question first:


  But how about receive rate? Can I reach higher receive rates? Or same with the 5dBi? 


Antennas are reciprocal. So, the same benefit you get on TX, you'd typically get on RX.

That was a simplification, because:


  If I change the antenna to 9 dBi, in the coverage zone with same range, I know I can reach higher transmit rate.


Well, in the main beam, you get 4 dB more signal power. If your 9dBi antenna really is that much better than your 5dBi antenna.

However, whether that translates to a higher data rate.

Data rate is certainly limited by the SNR at your receiver; which, at constant noise level, is defined by the amount of power reaching your receiver's antenna.

However, that is but one of many factors that limit rate. 

First of all, a receiver antenna with a high gain might also be getting more sensitive to an interferer that happens to also be in the main lobe now. Remember, you're in an ISM band, so everyone is allowed to communicate on your WiFi channel.

In a normal indoor scenario, it's not that the signal between your transmitter and your receiver only takes the direct path – it gets reflected on furniture and walls (especially steel armaments on those), and thus, you get a complicated multipath scenario. Classically, this is really a problem: Because these multiple paths have different lengths, the signals overlay, and might as well cancel out. Giving your transmitter a higher-gain antenna probably makes the shortest path better, but that doesn't mean things work better overall - it's very very hard to predict how the different path signals will overlap. And maybe, due to cancelling of the direct path with a path that is just half a wavelength longer, your receiver signal actually gets weaker because you use a higher-gain antenna.

Many modern access points and better WiFi adapters use what is called diversity gain with multiple antennas ("MIMO receivers"). Those actually benefit from the multiple paths – because each path has different properties, clever algorithms allow to get through more bit/s. But: for that you'll actually need the paths. In a bad-luck situation, your 4dB gain is actually decreasing your rate.

All in all, even if the additional 4dB directional gain actually just have the maximum benefit they can, i.e. they increase SNR by 4dB (\$=10^{\frac4{10}}\$), the harsh truth is that your expected (note: stochastic property!) increase in channel capacity can be bound by the ratio of Channel capacities with 5dBi and 9dBi antenna:

$$\begin{align}
\frac{C_{9dBi}}{C_{5dBi}} &amp;= \frac{B\log_2(1+\text{SNR}_{9dBi})}{B\log_2(1+\text{SNR}_{5dBi})}\\
&amp;= \frac{\log_2(1+\text{SNR}_{5dBi}\cdot 10^{0.4})}{\log_2(1+\text{SNR}_{5dBi})}\\
&amp;\approx \frac{\log_2(1+2.5\text{SNR}_{5dBi})}{\log_2(1+\text{SNR}_{5dBi})}\\
\text{for } 1 &amp;\ll \text{SNR}_{5dBi} \\
&amp;\approx \frac{\log_2(2.5\text{SNR}_{5dBi})}{\log_2(\text{SNR}_{5dBi})}\\
&amp;= \frac{\log_2(2.5) + \log_2(\text{SNR}_{5dBi})}{\log_2(\text{SNR}_{5dBi})}\\
&amp;= \frac{\log_2(2.5) }{\log_2(\text{SNR}_{5dBi})} + \frac{\log_2(\text{SNR}_{5dBi})}{\log_2(\text{SNR}_{5dBi})}\\
&amp;= \frac{\log_2(2.5) }{\log_2(\text{SNR}_{5dBi})} + 1\\
\end{align}$$

Which means that your 9dBi antenna gives you a rate increase by \$\frac{\log_2(2.5)}{\log_2(\text{SNR}_{5dBi})}\$.

Plug in "normal" SNRs for WiFi:

If your SNR now is \$15dB= 10^{1.5}\$, and with \$\log_2 2.5 = 1.3\$, your rate gain is a meagre \$\frac{\log_2(2.5)}{\log_2(10^{1.5})}\approx \frac{1.3}{5}\approx 26\%\$.

If your receiver SNR is \$30dB = 1000 \approx 2^{10}\$ now (so already pretty OK), your rate gain is only \$\frac{\log_2(2.5)}{\log_2(2^{10})}\approx \frac{1.3}{10}= 13\%\$.

All in all, unless you already have very weak reception, the 4dB won't do much good. And if you have really weak reception, either a really high-gain antenna will help (because you're at the wrong end of the block, for example), but you'd have to be careful not to break the law with that, or actually more antennas. So, getting a proper MIMO system with at least two  antennas on each end will very likely yield better results.

This is especially true because for WiFi, the problem usually isn't receiver noise (thus, SNR), it's interference on the ISM band (thus, SINR). And MIMO actually (almost) always can help with that, whereas it's really a random game whether a high gain antenna is the solution to an interference problem in a dense network. In fact, you can even make the so-called hidden-station problem worse by making one end of the communication link "deaf" to interferers that are very strong for the other end with a high-gain antenna. And such a problem typically isn't visible as signal strength metric, but leads to broken transmissions, and those are always far more expensive than having a slightly worse SNR.