This is just to store for future reference, but would a water well that has been dug make the property improved or would it still be considered unimproved?

In what context?

when it comes to LTV. Just wondering if a well that has been dug would make a piece of property improved or not.

I would not say that a well alone would be considered "improved". You normally need roads and utilities.

If the lot is not near city water lines, the well might be a "utility".

Would the hypothetical well have a pump and the pump a covering shed? That is improved, right?

Well, when all else fails, refer to the regulations:

Improved property loan means an extension of credit secured by one of the following types of real property:
(1) Farmland, ranchland or timberland committed to ongoing management and agricultural production;
(2) 1- to 4-family residential property that is not owner-occupied;
(3) Residential property containing five or more individual dwelling units;
(4) Completed commercial property; or
(5) Other income-producing property that has been completed and is available for occupancy and use, except income-producing owner-occupied 1- to 4-family residential property.