Learn Spanish (Latin America): Con relación al presupuesto, acaban de debatir los riesgos. - In relation to the budget, they have just debated the risks.

Yes, they all serve the same purpose here: to set the topic before making a statement about it. Con relación a (in relation to) and respecto a (regarding) sound slightly more formal or professional, making them perfect for a business context like discussing a presupuesto (budget). En cuanto a (as for) is equally correct but can feel a bit more everyday.

Why is it al presupuesto instead of a el presupuesto?

The topic-framing phrase is strictly con relación a. Whenever the preposition a is immediately followed by the masculine singular article el (since presupuesto is masculine), they must contract into al. You cannot leave them separated as a el.

Why use acaban de debatir for 'they have just debated' instead of the present perfect?

The structure acabar de + infinitive is the standard, foolproof way to express that an action just happened right before now. While you could say something like recién han debatido (or recién debatieron, which is very common in Latin America), acaban de debatir explicitly highlights the immediacy of the action.

Does the word presupuesto only apply to business contexts, or can I use it for my personal finances?