For some background on the various interrelated parts here and how they interact:
Webkit is the open-source browser engine used by Apple, and other companies, to power their web browsers and embed web content in other applications. It is used on all Apple platforms, to power the Safari browser and also to provide webview components in the API such as WKWebView and (deprecated) UIWebView. WebKit is a system library on Apple platforms because it is used to power those webview components. All browsers on iOS currently use the same WebKit system library. WebKit is maintained by the WebKit team at Apple, along with teams at companies like Igalia, Intel, Sony, and Red Hat, and open-source contributors.
Safari is Apple's web browser app, which provides a user interface on top of a WebKit-powered view and other common web-browser features such as bookmarks and tabbed browsing. Safari is closed-source, maintained by its own team at Apple. Because Safari is an Apple-internal project, it is allowed to use private WebKit APIs to provide closer system integration that are not available to other developers.
Web.app is the underlying implementation for what Apple calls "home screen web apps". It is separate from Safari (so it has its data storage) but also uses a WebKit-powered view. Because Web.app is an Apple-internal project, it is allowed to use private WebKit APIs to provide closer system integration that are not available to other developers. These private APIs are what are used to support things like push notifications and badging within Web.app, while not enabling those features in the Safari app.
Apple initially dropped support for Web.app on iOS 17.4 in the EU because it is powered by the system WebKit library, but is separate from a web browser application, and they've now backtracked that decision.
What the DMA enforced is that 3rd party web browser apps are no longer forced to use the system WebKit library for their web engine. So a Firefox app can ship with its Gecko engine, and a Chrome or Edge app can ship with its Blink engine. As well, apps that provide in-app browsing experiences (where links to web content open within the app instead of launching a browser app) are allowed to use a 3rd party web engine for rendering that content.
All other applications that embed web content within their application, must continue to use the system WebKit library through APIs with WKWebView. This includes "hybrid" apps (such as those made with tools like Apache Cordova or Ionic Capacitor), as well as apps that use web views to render portions of their content (such as the App Store).