Updating weights in Adaboost
weight for each node?  $w_i$

weight for each model? $\alpha_i$

table weight column (more confusing)? The initial weight is set as $\frac{1}{M}$ where M is the number of data points. I assume the data2_1 is the update of data1_1 after being updated by a new weight for feature1 but it's a bit obscure due to lack of description about the title, etc. of the table.

weight for each node is being updated after a classifier is trained on the data and it's weight ($\alpha_i$) and the score is calculated. 

$T(x_i)$ is the final classifier but seemingly is lacks a sign before the expression. which is the sum of multiplication of each classifier with it's weight $\alpha_i$