An Anglican High Church is named as such because of its formal and traditional beliefs and practices of ecclesiology, liturgy, and theology, which often overlap with Roman Catholicism.
But this has nothing to do with the "High" in "High Cross", other than such crosses are similarly ancient.
A High Cross is so named simply because it is tall.
Unlike the portable Processional Cross temporarily resting on a bracket in the church transept, a High Cross is usually an outdoor fixture.
The stem of a Processional Cross is typically made of wood, light enough to be carried in a procession. Surviving ancient examples of High Crosses are invariably made of stone, not only to withstand centuries of weather, but also because the broad stem would bear carvings of pictures or swirls. Such swirls are typically Celtic, and indeed, the top part of the piece tends to be a Celtic Cross.
They are found all over Britain and continue to be popular, as seen in the recently erected Saxon replica shown above, at Wakefield, England.
One of the oldest examples is found in Whithorn, a former royal burgh in Dumfries and Galloway, Scotland. This is the site of the 4th century Candida Casa, the first recorded Christian church in Scotland, and called the Monreith Cross (or Whithorn Cross).
The purpose for erecting such crosses was (and remains to be) as a focal point for an open-air service or meeting point, a memorial, to mark consecrated ground, which may be a burial ground, or simply a folly.
See Churchyard Cross for further information and examples.