There's a strict meaning and a loose one, and you've found the gap between them.
Strictly, value-add describes a return profile where a meaningful share of the total return comes from raising net operating income through work the owner performs, rather than from the income the property already produces. The classic test is whether the business plan requires physical or operational intervention to hit the projected NOI. Renovating units and re-leasing at higher rents is the standard case. The category sits between core-plus and opportunistic on the risk scale, and the industry uses those four buckets consistently enough that a consultant classifying a portfolio would put your second example in value-add without argument.
Loosely, sponsors apply the word to anything where they expect the price to be better later. Your first example is buying a discount to replacement cost and waiting for the supply wave to clear. That's a market-timing bet, and the NOI improvement comes from the submarket rather than from the sponsor. Your third example is a management story, which can be real (bad revenue management leaves money on the table) though it rarely justifies the return spread a value-add fund charges fees for.
Nothing forces a sponsor to use the term strictly. The PPM defines what it has to define legally and leaves marketing words undefined on purpose.
The place to look instead is the sources and uses table and the capital budget. A fund with real value-add has a renovation line, usually stated per unit, and a schedule of how many units get touched per month. If that budget is thin or absent, the word on the cover is describing hope about the market. Compare the projected rent premium per renovated unit to the per-unit spend and see whether the return on that spend is credible on its own.
