It seems that your (4) refers to the decomposition of $su(3)$ irreps whereas your (3) applies to the decomposition of $su(2)$ irreps.
The overline refers to the $su(3)$ irrep conjugate to $3$ ($\bar{3}$would be the antiquark irrep). Whereas all irreps of $su(2)$ are self-conjugate so in $su(2)$ there is no difference between $3$ and $\bar 3$, conjugate irreps in $su(3)$ are different - so $\bar 3$ and $3$ are different Indeed in $su(3)$ the tensor product $3\otimes 3=6+\bar 3$,
where $6$ is the symmetric part of the tensor product and $\bar 3$ is the antisymmetric part.
Note that HEP people has the sad habit of referring to irreps by their dimension: whereas $3$ and $\bar 3$ are distinct, they have the same dimension hence the bar. In the more mathematical literature, where irreps are distinguished by Dynkin labels, $3$ is $(1,0)$ and
$\bar 3$ is $(0,1)$, while $6$ is actually $(2,0)$. Thus:
$3\otimes 3\to 6\oplus \bar 3$ is written $(1,0)\otimes (1,0)\to (2,0)\oplus (0,1)$.
The Littlewood Richardson rule applies for $su(3)$ and $su(2)$ decomposition however, whereas the $su(2)$ irrep $j=3/2$ would be represented by a Young diagram with $3$ boxes, the $su(3)$ irrep $3$ is represented by a Young diagram with a single box, and $\bar 3$ by a Young diagram with two boxes in a single column of two rows.
If you prefer the notation in terms of partitions, then $3$ corresponds to the partition $\{1\}$ but
$\bar 3$ to the partition $\{1,1\}$. $su(2)$ irreps correspond to partitions with a single row of length $2j$.
Details of applications of the Littlewood-Richardson rules can be found in very many places (see here and here for a start)
Edit:
The dimension of the $su(3)$ irrep $(\lambda,\mu)$ (in Dynkin notation) is
$$
\hbox{dim}(\lambda,\mu)=\frac{1}{2}(\lambda+1)(\mu+1)(\lambda+\mu+2)
$$
so with this
$$
\hbox{dim}(1,0)=\frac{1}{2} 2\times 1 \times 3=3
$$
and so forth. The difficulty of course is that there might be more than one pair of $(\lambda,\mu)$ which produce the same dimension: indeed $(\mu,\lambda)$ has the same dimension as $(\lambda,\mu)$. In this case, one of the irrep is assigned a $*$ as $(\lambda,\mu)$ and $(\mu,\lambda)$ are conjugate: the eigenvalues of the Cartan elements in one are the negatives of the eigenvalues in the other. It is possible there are more than two irreps with the same dimension, in which case a double $**$ is used. I do not know of a systematic convention to assign $*$ or $**$ but the standard go-to text for this is the review by Slansky.
The Dynkin labels $(\lambda,\mu)$ are the eigenvalues of the Cartan elements on the so-called highest weight state.
The Lie algebra $u(n)$ is spanned by operators $C_{ij}$ with commutation relations
$[C_{ij},C_{km}]=C_{im}\delta_{jk}-C_{kj}\delta_{im}$. This is realized by matrices
$$
C_{ij}=\cases{
1& at row $i$ column $j$\\
0& otherwise} \qquad \hbox{for} i\ne j. \tag{1}
$$
The Cartan elements are diagonal with
$$
h_k=\cases{1& at row $k$ and column $k$\\
-1 & at row $k+1$ and column $k+1$\\
0& otherwise}\, . \tag{2}
$$
Note that these differ from the standard diagonal Gell-Mann matrices. With this, $C_{ij}$ with $i<j$ is a raising operator and the highest weight state of a representation is the unique state killed by all raising operators, the eigenvalues of $h_k$ are its weight and the Dynkin labels are just these weights, which can be shown to be non-negative integers. With this convention the Dynkin label for $su(2)$ irreps should be $2j$ (which is necessarily an integer).
This definition also makes it obvious that the $N$-dimensional vector
$(1,0,\ldots)$ is a highest weight for the $su(N)$ irrep $(1,0,\ldots)$. The other $N-1$ vectors span the $N$-dimensional space and the matrices resulting from (1) and (2) are used to define the fundamental representation of $su(N)$.
