I fitted a Bayesian model and the histogram below is from a posterior estimation.
I calculated the empirical probability that $\beta_3$ is bigger than $0$. This probability is $P(\beta_3>0)\approx 0.94$ is wrong if I say that this parameter is non-zero?
You have calculated two probabilities about $\beta_3$: 1) the 95% HPD includes zero and below, and 2) you calculate that P($\beta_3 > 0$) is about 94%. If you were writing a paper, you could use either of these textual descriptions or better yet show the graph. It would depend on your field and audience as to whether you could draw a conclusion that $\beta_3$ was likely to be strictly positive or not.
The 95% threshold is arbitrary, of course, and some fields set the threshold at 95%, but I've seen 50%, 80%, 95%, and 99% used.
Also, you have not set a region around zero that you would say is equivalent to zero, what Kruschke calls your ROPE (Region of Practical Equivalence, I believe) for zero. If, in your field and measurements, results within 0.1 of a value are essentially equivalent to that value, then your ROPE for zero is [-0.1, 0.1] and your posterior doesn't come close to excluding that.
If your ROPE contains 95% of your posterior -- and 95% is your threshold -- you could say that $\beta_3$ is zero. If 95% of your posterior excludes your ROPE, you could say that $\beta_3$ is non-zero. Otherwise, you must say that there is not enough evidence to say one way or the other. (This is an improvement over frequentist statistics which can only reject, never accept.)
(This gets into the important distinction between statistical significance versus practical significance.)
Thinking of it as zero or non-zero is wrong. The idea that it is zero is important in null hypothesis methods, but problematic in Bayesian methods. This is because $\Pr(\beta_3=k)=0$. In this case $k=0$, but it doesn't matter what $k$ is. The measure of a single point in a continuum is zero so its probability is zero. Intuitively, think of it as $\lim_{n\to\infty}\frac{1}{n}=0$. You have an infinite number of points and you want to put a hypothesis that it is EXACTLY one point. You cannot do that in any reasonable way. In the subjective Bayesian interpretation $\beta_3$ is never a fixed point anyway so speaking of it as equal to something means nothing. Rather, nature draws it from a distribution.
Whether or not $\beta_3$ is "significant" isn't found by seeing if it is equal to zero, but rather by determining if models without it have higher posterior probabilities than models with it. Consider the model $$y=\beta_1x_1+\beta_2x_2+\beta_3x_3+\alpha.$$
There are multiple possible hypotheses. It could be that $$y=\alpha,\text{ or}$$ $$y=\beta_1x_1+\alpha,\text{ or}$$ $$y=\beta_2x_2+\alpha,\text{ or}$$ $$y=\beta_3x_3+\alpha,\text{ or}$$ $$y=\beta_1x_1+\beta_2x_2+\alpha,\text{ or}$$ $$y=\beta_1x_1+\beta_3x_3+\alpha,\text{ or}$$ $$y=\beta_2x_2+\beta_3x_3+\alpha,\text{ or}$$ $$y=\beta_1x_1+\beta_2x_2+\beta_3x_3+\alpha.$$ You would calculate the posterior probability that each of these models are the true model. Then you could choose Bayesian model selection, Bayesian model averaging or you could sum the posterior probabilities that include $\beta_3$ versus models without it.
As mentioned above, there is a different practical question of scientific usefulness, which is what ROPE gets at.
There is a nice video on this I came across at https://www.youtube.com/watch?v=w6AjduOEN2k&feature=youtu.be
