Inequality convention in upcrossing definition

Question

My notes define an upcrossing of $[a,b]\subset \mathbb{R}$ using strict inequalities, i.e. (roughly) an index interval $\{j,j+1,\dotsc,k\}$ such that $$X_j<a, \quad \text{and}\quad X_k>b$$ However, I've seen this defined using non-strict inequalities as well. Is one convention more preferable to the other? Also, it looks like the former alternative gives fewer upcrossings than the latter, which I guess would make e.g. Doob's upcrossing inequality a weaker statement?