[1][2] A smoother is an algorithm that implements a solution to this problem, typically based on recursive Bayesian estimation.
Without a backward pass (for retrodiction), the sequence of predictions in an online filtering algorithm does not look smooth.
The historical reason for this confusion is that initially, the Wiener's suggested a "smoothing" filter that was just a convolution.
Later on his proposed solutions for obtaining a smoother estimation separate developments as two distinct concepts.
These names are used in the context of World War 2 with problems framed by people like Norbert Wiener.
This is used in the context of World War 2 defined by people like Norbert Wiener, in (stochastic) control theory, radar, signal detection, tracking, etc.
[1][2] Most importantly, in the Filtering problem (sense 2) the information from observation up to the time of the current sample is used.
Filtering is the estimation of a (hidden) time-series process based on serial incremental observations.