You are correct. In order for this to be a pure fx carry trade with no market risk (just credit risk), you have to have 2 fx trades. One shorter dated and the other longer dated.
Typically, you this will be done where you buy a currency spot and sell that same amount of currency in the future, in your case through an FX future. The difference between where you bought the currency and where you sold it in the future would be the carry that you earn.
As an example, say you bought 1MM USD of JPY at spot 115.89, you will pay 1MM USD and receive JPY 115,890,000. And through a fx future, you sold that JPY 115,890,000 on Jun 15 at 115.6652, you will receive USD 1,001,943.54. You will have earned USD 1,943.54 carry from now until Jun 15.
A few of things to consider however is 1) you will have to maintain margin for your futures position which will cut into your carry; 2) with negative interest rates, you will be paying interest on the spot currency you own (eg EUR, JPY etc). Your bank will charge you for having a balance in these currencies (again, cutting into the carry you will earn) and; 3) Futures are standardized contracts and you will have to buy and sell in increments of the notional of the future.
Edit for your edited question: In your example from the CME paper, the investor is short USD, long AUD at a future date. The investor has agreed to purchase AUD (paying in USD) at the agreed upon price in the future when they entered the contract. As such they are now long AUD and have market risk to USD/AUD exchange rate. Provided they still hold the future at maturity, the exchange will assign the trade to them and they will have to deliver the USD they agreed to pay and receive the AUD they agreed to buy.
If this exchange rate had moved unfavorably for the investor by the time to settle the future, they will be buying AUD at a price that is higher than the spot. Should they want to take the AUD risk off at that point they would convert the AUD to USD at spot and lose money on the trade. (Of course the spot exchange rate could have moved into their favor and they would have been buying AUD cheaper).
Futures contracts embed the term rates for the underlying currencies into the contract price (the price that the investor will agree to purchase the AUD in the future via the futures contract). One way to visualize the this: (1) if an investor would need to deliver USD in the future for the AUD, how much would they have to set aside now to make good on that obligation? The amount that the investor would set aside would be earning the USD term interest rate to the future date and (2) How much interest would this investor forego by agreeing to take the AUD in the future rather than now? Had they purchased AUD now, they would have earned the AUD term interest rate to the future date.