I want to use some examples to comprehension the definition of free product. Let $\mathbb{Z_{2}}$ be the integers $\{o, ...,m-1\}$ with addition modulo $m$ as the group operation and $\Gamma=\mathbb{Z_{2}}*\mathbb{Z_{2}}*\mathbb{Z_{2}}$ be the 3-fold free product of $\mathbb{Z_{2}}$. What are the reduced words in $\Gamma$?
I use $a, b, c$ to denote the non-identity element in the first $\mathbb{Z_{2}}$, the second $\mathbb{Z_{2}}$ and the third $\mathbb{Z_{2}}$ (Indeed, $a=b=c=1\in \mathbb{Z_{2}}$). And I think the reduced word is that it does not contain the same letter in two neighboring positions. Right？ And the $\{a, b, c\}$ are the generators of $\mathbb{Z_{2}}*\mathbb{Z_{2}}*\mathbb{Z_{2}}$, right?
Let us set $G_i$ for the $i$-th copy of $\mathbb{Z}_2$ ($i=1,2,3$).
The words in $\Gamma$ are (by definition) :
$$a_1...a_r\text{ where } a_k\in G_1\cup G_2\cup G_3 $$
Now if some $a_k$ is $0$ in $G_i$ then you can delete it (via reduction) so in fact if we set :
$$G_i:=\{0_i,1_i\}$$
A reduced word should be of this form :
$$a_1...a_r\text{ where } a_k=1_i \text{ with }i=1\text{, }2 \text{ or } 3 $$
Finally to make sure that it is indeed a reduced word you must add the following condition :
$$\text{ If } a_k=1_i\text{ and } a_{k+1}=1_j\text{ then } i\neq j $$
This means that two adjacent letters cannot be in the same copy of $\mathbb{Z}_2$ (the same thing you said).
Finally :
$$\Gamma=\{\epsilon\}\cup\{1_{i_1}1_{i_2}...1_{i_k}|i_l\neq i_{l+1}\}$$
In my notation we see that $1_1$, $1_2$ and $1_3$ generate $\Gamma$ so indeed, with your notation $\{a,b,c\}$ is a generating set of $\Gamma$.