Learn Italian: Questa immagine mostra il centro della città. - This picture shows the centre of the city.

The rule you are currently learning applies specifically to appositions—descriptive phrases placed right next to a proper noun (like Roma, capitale d'Italia). In this sentence, il centro isn't an apposition defining a proper noun; it is simply the direct object of the verb mostra, so it requires its standard definite article.

Why use mostra here instead of dimostra?

Mostrare means "to show" or "to display" visually, which is exactly what a picture does. Dimostrare means "to demonstrate" or "to prove" an argument. Unless the picture is being used as logical evidence to prove a specific point, mostra is the natural choice.

Could I write quest'immagine with an apostrophe instead of questa immagine?

Yes, both questa immagine and quest'immagine are perfectly correct. Dropping the final -a and adding an apostrophe before a vowel is very common in everyday Italian, while keeping the full word (questa immagine) is slightly more formal but equally common.

Could we say il centro città instead of il centro della città?