Is it a work phone because Trump has a principled belief that only work phones should have to install this app? Or is it a work phone because Trump hasn't yet subverted the required executives to get Apple to agree to roll it out on everyone's phone?

"get them to agree" is one way to put it. If they want this our government can just show up with guns and force it to happen. The government already collects data from corporations (including apple according to snowden's leaks), sometimes taking over parts of a company's offices, installing hardware on the company's network, and filling those rooms with government employees in order to do it. https://en.wikipedia.org/wiki/Room_641A