The terminology I have seen in the literature refers to such a sequence as being a *$1$-suppression unconditional basis*. More generally, if for $K\geq1$ we have \begin{equation}\Vert \sum_{i\in F}x_ie_i\Vert \leq K\Vert x\Vert\end{equation} for every $x=\sum_{i=1}^\infty x_i\in X$ and finite $F\subset\mathbb{N}$, then we say that $(e_i)_{i=1}^\infty$ is *$K$-supression unconditional*. The least constant $K$ for which the above inequality holds is called the *suppression constant* of $(e_i)_{i=1}^\infty$, which is sometimes denoted by $K_s$. (If $K_u$ is the unconditional basis constant of $(e_i)_{i=1}^\infty$, then clearly $1\leq K_s\leq K_u\leq 2K_s$.)

As far as references go, most of the above is contained in Remark 3.1.5 of *Topics in Banach Space Theory* by Fernando Albiac and Nigel Kalton. See also the paper *Characterization of 1-almost greedy bases* by Albiac and Ansorena (doi: 10.1007/s13163-016-0204-3, arXiv:1506.03397).