Were the women—with Mary Magdalene—saying “Jesus appeared to us”?

The gospels report that Jesus predicted his death and resurrection. Days after dying on the cross, Mt and Lk report that Jesus appeared some female disciples—including Mary Magdalene—as if Jesus was alive again. Were the women and/or Mary ultimately the source of this report? Was Mary Magdalene and other named women confidently maintaining that Jesus appeared to them near Jesus’s tomb when they visited it? Did she/they publicly maintain that this was their experience? (Did this occur at the tomb on Easter, c. AD 30).

Craig Keener: “Most thus judge the report that the women were the first witnesses of the resurrection message to be historically accurate.” [The Historical Jesus of the Gospels (Eerdmans, 2009), 331.]
Gerald O'Collins & Daniel Kendall: “One who accepts this tradition [that Jesus appeared alive after death] should agree that women were the first or among the first witnesses. In the tradition women, especially Mary Magdalene, have a lead role. Above all, in John 20 Mary Magdalene is the human figure who holds the events together. [“Mary Magdalene as Major Witness to Jesus’ Resurrection” Theological Studies 48 (1987), 645.]

Though some are skeptical of the actual narration:

Gerd Lüdemann: “It seems to be historically certain that Mary Magdalene experienced an appearance of the risen Jesus. … But the tradition [i.e. the pop-story itself which circulated among communities of Christians] of a first appearance to Mary arose relatively late (see above, 64ff.), whereas that to Peter is among the earliest pieces of tradition. This is a good reason for regarding the first appearance [i.e. the full-blown tradition story] to Mary as unhistorical.” [*What really happened to Jesus*, Trans. by Bowden (Westminster, 1995), 80-81.]

See the related section on whether the women witnessed Jesus's tomb empty.