Through Time With Mark Twain, HG Wells And Their Successors...

...mainly including Poul Anderson, of course.

The anonymous "Missing One's Coach" presents a visit to a historical period that might be a dream.

Mark Twain's Connecticut Yankee experiences instantaneous "transposition of epochs" to Arthurian Britain and returns to the nineteenth century by Merlinian suspended animation.

In The Time Machine by HG Wells, the Time Traveler and his dinner guests discuss "time travelling" before the Time Traveler demonstrates his model Time Machine which does not move anywhere but, remaining stationary on the Earth's surface, undergoes extreme time dilation in either temporal direction while also unaccountably becoming both invisible and intangible.

In "The Dark Tower" by CS Lewis, five men discuss "time travelling" and agree that it is impossible before one of their number demonstrates his chronoscope which displays scenes not, as they initially think, from the past or future but from an alternative timeline.

In the Time Patrol series by Poul Anderson, timecycles, resembling modenized Time Machines, instantaneously change their spatiotemporal coordinates, thus neither moving nor dilating but disappearing and appearing.

In The Corridors Of Time by Poul Anderson, "time travelers" really do travel through time because they walk or drive down corridors whose lengths extend along the temporal axis.

In The Dancer From Atlantis by Anderson, a space-time vehicle does not remain stationary on the Earth's surface but moves across it "...while traveling through time." (London, 1977, p. 32)

In There Will Be Time by Anderson, some mutants can "time travel"/dilate invisibly and intangibly without needing a time machine and one of their number gives Wells the time travel idea.

In The Time Traveler's Wife by Audrey Niffenegger, mutant time travel is instantaneous but involuntary.

In Bid Time Return by Richard Matheson and two Time novels by Jack Finney, travel to the past and back to the present is a learned skill.

In "The Flight of the Horse" by Larry Niven, time travel is scientifically impossible so it takes explorers into a fantasy past where they find not extinct species but mythical beasts.

This is not and cannot be a complete list of time travel fictions but it does cover most of the possibilities and presents a conceptual sequence:

time travel is possible with machines - but there are different kinds of machines;
time travel is possible without machines - but in different circumstances;
time travel is impossible but time viewing is possible with a machine;
time travel is impossible so it is fantasy, not sf.