Can someone give me a very precise answer about what to enter in “improvement value”? Is it the total purchase value minus land value plus costs of improvements?

Precisely where are you being asked to enter an "improvement value"?

I hope TurboTax corrects their instructions in the software because it leads to a completely different number than your advice.

There's really nothing wrong with it, and there is no "perfectly" correct way to state it. Some tax bills will call it land value and structure value. Others will call it Total Value and Improvement value (where improvement value is that portion of the total value given to the structure) and still others will call it total value and land value (where the difference is the structure value). So there is no "perfect" way to word it that will make everyone happy. Just flat out not possible.

Also, as a first time landlord (or first time using TurboTax as a landlord) here's some clarifications you will need, that the program (in my personal opinion) doesn't provide. Take special note that "absolute" "perfection" is not an option in the first year of using TurboTax with rental property. It's an absolute *must*. Even the tiniest of mistakes *will* grow exponentially over time. Then when you catch it later down the road (if the IRS doesn't catch it first) the cost of fixing it will be quite high.

I also recommend that you do "not" include any property improvements that you did after your initial purchase of the property, in your cost basis of the actual property itself. (but you can if you want.) When you add your property improvements to the cost basis in the "initial" entry for the property itself, the program incorrectly assigns a portion of the cost of that improvement to the land. So for any property improvements you did "after" you purchased the property, enter those improvements as a separate asset in the Assets/Depreciation section.

Rental Property Dates & Numbers That Matter.

Date of Conversion - If this was your primary residence or 2^nd home before, then this date is the day AFTER you moved out, or the date you decided to lease the property – whichever is later.
In Service Date - This is the date a renter "could" have moved in. Usually, this date is the day you put the FOR RENT sign in the front yard.
Number of days Rented - the day count for this starts from the first day a renter "could" have moved in. That should be your "in service" date if you were asked for that. Vacant periods between renters count also PROVIDED you did not live in the house for one single day during said period of vacancy.
Days of Personal Use - This number will be a big fat ZERO. Read the screen. It's asking for the number of days you lived in the property AFTER you converted it to a rental. I seriously doubt (though it is possible) that you lived in the house (or space, if renting a part of your home) as your primary residence or 2nd home, after you converted it to a rental.
Business Use Percentage. 100%. I'll put that in words so there's no doubt I didn't make a typo here. One Hundred Percent. After you converted this property or space to rental use, it was one hundred percent business use. What you used it for prior to the date of conversion doesn't count.

RENTAL PROPERTY ASSETS, MAINTENANCE/CLEANING/REPAIRS DEFINED

Property Improvement.

Property improvements are expenses you incur that “better” the property. Basically, they retain or add value to the property. Expenses for this are entered in the Assets/Depreciation section and depreciated over time. Property improvements can be done at any time after your initial purchase of the property. It does not matter if it was your residence or a rental at the time of the improvement. It still adds value to the property.

To be classified as a property improvement, two criteria must be met:

1) The improvement must become "a material part of" the property. For example, remodeling the bathroom, new cabinets or appliances in the kitchen. New carpet. Replacing that old Central Air unit.

2) The improvement must retain or add "real" value to the property. In other words, when the property is appraised by a qualified, certified, licensed property appraiser, he will appraise it at a higher value, than he would have without the improvements.

There are rules that allow you to just flat-out expense and deduct some property improvements, if the total cost of the improvement was less than $2,500. It’s referred to as “safe harbor di-minimis” But depending on the specific situation, this may or may not be beneficial. Just be aware that not every property improvement that cost less than $2,500 qualifies for this. If this interest you, the rules can get complex. So a good place to start reading is on the IRS website at https://www.irs.gov/businesses/small-businesses-self-employed/tangible-property-final-regulations. The stuff on di-minimis starts about one page down.

Cleaning & Maintenance

Those expenses incurred to maintain the rental property and it's assets in the useable condition the property and/or asset was designed and intended for. Routine cleaning and maintenance expenses are only deductible if they are incurred while the property is classified as a rental. Cleaning and maintenance expenses incurred in the process of preparing the property for rent are not deductible.

Repair

Those expenses incurred to return the property or it's assets to the same useable condition they were in, prior to the event that caused the property or asset to be unusable. Repair expenses incurred are only deductible if incurred while the property is classified as a rental. Repair costs incurred in the process of preparing the property for rent are not deductible.

Additional clarifications: Painting a room does not qualify as a property improvement. While the paint does become “a material part of” the property, from the perspective of a property appraiser, it doesn’t add “real value” to the property.

However, when you do something like convert the garage into a 3^rd bedroom for example, making a  2 bedroom house into a 3 bedroom house adds “real value”. Of course, when you convert the garage to a bedroom, you’re going to paint it. But you will include the cost of painting as a part of the property improvement – not an expense separate from it.