Let $X_1, \dots, X_n$ be random variables, for some integer $n > 2$. Then, $$\text{Var}\left(\sum_{i=1}^nX_i\right) = \sum_{i=1}^n \text{Var}(X_i) + \sum_{i \neq j} \text{Cov}(X_i, X_j) $$
There is a natural interest in the special case when the $X_i$ are such that $$\sum_{i \neq j} \text{Cov}(X_i, X_j) = 0 \quad \Leftrightarrow \quad \text{Var}\left(\sum_{i=1}^nX_i\right) = \sum_{i=1}^n \text{Var}(X_i)$$
Now, the condition $$\text{Cov}(X_i, X_j) = 0, \,\forall\,i \neq j\tag{1}$$ clearly implies the condition $$\sum_{i\neq j} \text{Cov}(X_i, X_j) = 0 \;.\tag{2}$$
Is the converse true?
Alternatively, can someone give me an example where $(2)$ holds but $(1)$ fails?
(My guess is that $(2)$ does not imply $(1)$, since I realize that, in general, $\sum_1^{n>1} a_i = 0$ does not imply that all the $a_i$ are zero, but I can't readily come up with an example of this general fact when the summands are the terms $\text{Cov}(X_i, X_j),\;(i\neq j)$.)
(My remaining questions are moot if, contrary to what I suspect, $(2)$ does imply, and is therefore equivalent to, $(1)$.)
When the random variables $X_1, \dots, X_n$ are described as being "uncorrelated", does this mean that they satisfy $(1)$, or just that they satisfy $(2)$?
Lastly, I've often seen the term "pairwise uncorrelated", which clearly means that $(1)$ holds, is there a name for the condition in $(2)$ (when $(1)$ is not assumed)?