my client only paid half the invoice, how do I record that

Record what they actually paid, not what you invoiced. The invoice stays open for the rest.

Open the invoice and record a payment for the amount received, dated the day the money landed. The invoice moves to partly paid, its remaining balance keeps showing in what you're owed, and it stays on your aging report until the rest arrives. When the balance comes in, record a second payment against the same invoice and it closes.

Why not just edit the invoice down to what they paid

Because then you've quietly written off the difference and lost the fact that you're still owed it. The invoice is the agreement; payments are events against it. Keeping them separate is what lets you answer "who still owes me, and how much" at any point, and it means the receivable on your balance sheet is real.

If the shortfall is genuinely never coming (a discount you agreed, or a bad debt) that's a different action from a partial payment, and worth handling as its own decision rather than by shrinking the invoice.