UCT1 Algorithm: What does "total number of simulations" mean?
The UCT1 algorithm is actually an algorithm for a multi-armed bandit. There is a machine with several arms. At each round you pull one of the arms and get some reward. Your goal is to maximize your total reward. In this algorithm, $t$ is the round number: $t = 1$ in the first round, $t = 2$ in the second round, and so on.

When using UCT1 to perform Monte Carlo tree search, you treat each explored node as a multi-armed bandit. Monte Carlo tree search consists of several rounds of simulations: in each round a complete game is played out. The value of $t$ for a particular node is the number of rounds at which you passed through the node.

$n_i$ is the number of simulations through the $i$th child, i.e. how many simulations encountered that state in their simulations. A simulation consists of an entire game played out. Once it exits the node it invariably goes through one of its children. So $t$ is the sum of the $n_i$.

You ask:


  I made some images. Is it like this or like this?


It's more like the latter.