In this paper, we propose a new notion of strong networked detectability of networked discrete-event systems (DESs) modeled by finite-state automata (FSAs) with respect to a first-in, first-out channel with communication losses and bounded transmission delays. We use a new method to obtain a polynomial-time algorithm for verifying strong networked detectability. Moreover, we study the relationship between the new notion and its counterpart studied by [Sasi and Lin 2018]. We prove that strong networked detectability is strictly weaker than its counterpart in [Sasi and Lin 2018] and also obtain a polynomial-time verification algorithm for its counterpart without any assumption, which is more efficient than the exponential-time algorithm given by [Sasi and Lin 2018] under several assumptions. Particularly, for FSAs with respect to an ideal channel, under two fundamental assumptions of deadlock-freeness and having no unobservable reachable cycle, we show that strong networked detectability is equivalent to its counterpart studied by [Sasi and Lin 2018].