The Mold Earth Cantrip
Funny enough, I've actually had a similar discussion with my 5th Edition D&D group. My search for the difference between dirt and ground led me to this question, which led me to the look at the wording of the Mold Earth cantrip:
You choose a portion of dirt or stone that you can see within range and that fits within a 5-foot cube. You manipulate it in one of the following ways:
• If you target an area of loose earth, you can instantaneously excavate it, move it along the ground...
• You cause shapes, colors, or both to appear on the dirt or stone...
• If the dirt or stone you target is on the ground, you cause it to become difficult terrain. Alternatively, you can cause the ground to become normal terrain if it is already difficult terrain...
Interpreting the Wording of the Mold Earth Cantrip
- "If you target an area of loose earth, you can instantaneously
excavate it, move it along the ground ..."
I believe loose earth means "broken up" earth: the substance of the land surface; soil (soil, clay, loam, etc.) - from the 2nd definition of "earth" provided by Google.
Here, loose earth describes the substance being moved and ground describes the terrain across which it is being moved: the he terms are not interchangeable. We may extrapolate that "ground" refers broadly to "any solid surface on the planet", including stone or ice not covered by soil, per the Google's definition of "ground".
- "If the dirt or stone you target is on the ground, you cause it to become difficult terrain. Alternatively, you can cause the ground to become normal terrain if it is already difficult terrain."
Here, a distinction is made between "dirt" / "stone" and "ground". The use of the conjunction if indicates that the dirt or stone is not necessarily one and the same as the ground. This supports my claim that, according to the 5th edition rules of D&D, the term "ground" is not interchangeable with "loose earth", "soil", "dirt", "stone", and so on.
Relationship of Terrain and Movement
Lastly, I think it is important to clarify the definition of difficult terrain in 5th edition D&D:
... adventurers often face dense forests, deep swamps, rubble-filled ruins, steep mountains, and ice-covered ground - all considered difficult terrain.
- 5th edition D&D, Player's Handbook (pg. 182)
In this excerpt and in the Mold Earth cantrip's description, the term "ground" is used in relation to the terrain. Here, ground very clearly is used to describe the solid surface while the term terrain is used when discussing movement on that ground.
A Counter Argument: The Move Earth Spell
... Because the terrain's transformation occurs slowly, creatures in the area can't usually be trapped or injured by the ground's movement.
- 5th edition D&D, Player's Handbook (pg. 263)
Here, "terrain" and "ground" are used interchangeably to describe a solid surface (where the terrain's transformation causes the ground's movement). Regardless, this doesn't confuse the concept of ground... I only bring this up to remind you that, depending on the context, "terrain" may refer to 1) describe the solid surface or 2) discuss movement upon that surface.
My Final Answer
Pulling it all together...
Does the ground simply mean whatever solid surface the characters and monsters are standing on?
Yes. The ground is a surface that could conceivably harbor some object on it. When discussing motion on that surface (of characters, monsters, bookshelves, etc.), most text in 5th Edition D&D rule-books uses "terrain". Sometimes, "terrain" is used instead of "ground" (but not the other way around).
Again... ground refers broadly to any solid continuous, contiguous surface on the planet. Terrain refers broadly to any solid continuous, contiguous surface that the characters or monsters move on.
Terms like dirt, stone, and ice are very specific and can only be interpreted as dirt and stone, respectively - they are not interchangeable. According to Jeremy Crawford's tweet, dirt and loose earth are fairly interchangeable terms:
Q: What's loose earth for the Mold Earth cantrip? Not stone, or a more specific type of dirt structure?
A: Think dirt, not stone.
As a GM, I'd clearly define "dirt" separately/distinctly from "soil", (loose earth and dirt can be interchangeable). Dirt isn't necessarily arable, while soil is necessarily arable.
As a note, I have heard people say "fertile soil" (never "fertile dirt"). As @GreySage commented, this may mean that soil may be "infertile". I'd imagine that in such a case, you'd simply refer to it as dirt, sand, etc. To be on the safe side, take care as a GM to specify the ground's soil type when needed so that the ground is "well-defined" for players to choose spells and make role-playing decisions.
For reference:
Soil: the upper layer of earth in which plants grow, a black or dark brown material typically consisting of a mixture of organic remains, clay, and rock particles.
Dirt: earth used to make a surface for a road, floor, or other area of ground.
Best Answer
The hand does persist
A successful save only changes whether/how the target is affected by the spell
In fact, every spell persists for the duration even after a successful save unless there is specific mention of the spell ending.
This is because the saving throws only allow targets to avoid effects of the spell rather than cancel the spell as a whole.
How this works with other spells
For example, charm person will last the whole hour unless it is dispelled, even if the target succeeds on the save (although the spell then has no effect) since there is no mention of the spell ending otherwise.