The most common reason for Diawi to show "Download application" and not "Install application" is Safari being in "Desktop mode".

This mode is most likely enabled in the Settings app > Safari > Desktop version where it can be configured for some websites or all. It can also be switched on/off for a single website using the url bar's "Aa" or "square with two lines underneath" buttons.

When this mode is enabled, Safari on the iPhone sends a Safari Desktop user agent to the website, and this may cause some issues with many websites.