Deleted app from Home Screen

depending on your settings, downloading the app again may just place it back in the App Library.. so to bring this full circle, you should learn how to move apps from the App Library back to the Home Screen

From your Home Screen, swipe left until you get to the App Library.

Swipe downward and you'll get an alphabetical list of your apps.

Locate the app in the list or type the app name into the search field

Once you've located the app, tap/hold until it raises/selects, then drag it and it'll pop out to your Home Screen

If you don't like the alphabetical list and prefer the categories.. locate the app icon, tap/hold and select Add to Home Screen

Organize the Home Screen and App Library on your iPhone - Apple Support

Well, there may be a misunderstanding: swiping left means swiping from the right to the left. Meaning that the App Library will be the last page on your phone, after swiping left several times (depending on the number of pages you have.)

Swiping right, i.e. from the left to the right, will display the Search screen just like you mentioned.

Now, from The App Library, you will be able to delete an app by long-pressing it and tapping Delete.