I commonly use ArcMap with a Lambert Conformal, NAD83 HARN CRS. When adding the ESRI imagery map service (either from the Add Data > Basemap option, or from the Add Data from ArcGIS Online option) the imagery displays, followed quickly by the Geographic Coordinates System Warning window:
I understand that a datum shift is required, so I dutifully click the Transformations button, select the appropriate transformation, click Ok, and continue mapping. So far, so good.
Over the years, however, I've noticed that no matter which transformation I select, the ESRI imagery never shifts (remember that the imagery already exists, because it appears before the Warning window). Furthermore, even if I don't click the Transformations button, and simply click Ok, the ESRI imagery does not shift, even when zoomed in.
In other words, the position and shape of the initial ESRI imagery does not change, no matter which options are selected in the Warning window. I've asked a few GIS associates about this behavior, and they all replied that they never clicked the Transformations button, instead simply clicking Ok. When pressed, they all said that it didn't make any difference, which seems to validate my experience.
But this has got me curious: why have the Warning window at all if none of its options make any difference? What happens internally if I simply click the Ok button without selecting a transformation? The ESRI help section is silent on this issue.