Oil threads can be very opinionated about what brand. Many brands will work fine so I try to never say one brand is best since I don't believe any one brand IS best in all situations. You can use what you want as far as brand or viscosity within the limits of certain factors. 10W-30, 10W-40, 5W-40, doesn't matter. They will ALL be the right oil for you if you don't exceed the low temps ratings as given in that graphic above. You can use Full Synthetic, Synthetic Blend, or full dino oil. Most Synthetic oil is made using dino oil anyway but modified to not break down as fast.
@LaGrasta 's chart is the same as what is shown in the service manual for the CTX1300 so you can use any that is shown there. These are guidelines, not hard and factual limits. So as long as the low temp is paid attention to you'll have the right oil. All the viscosity ratings listed work at the high temp limits and it is unlikely you'll be riding so far over 100 F that you have to worry about anything different. And don't be too concerned if the first and last numbers of the viscosity don't exactly match the pairs or sets of viscosity in that chart as long as each number doesn't exceed those shown. Variable viscosity oils will act like the first number before you start the engine so it needs to be lower to more easily flow to and coat moving parts. Then as the engine warms up the oil viscosity changes toward the second number until full operating temp is reached and the oil will then be at the viscosity of the second number.
That said there are limiting factors that can definitely damage the engine, and more correctly the clutch. That being if the oil is designated as Resource Conserving or Energy Conserving. Those types of oils have special anti-friction additives or modifiers that will damage the clutch plates in a bike that shares the engine oil with the clutch and cause the clutch plates to slip. That is called a wet clutch and the CTX1300 has a wet clutch. There are additives in almost all oils that you may use but some have special modifiers that you want to stay away from for wet clutch motorcycle applications. That designation of Resource Conserving or Energy Conserving is shown in the API Service circle in the outer bottom circle. If that outer bottom circle is blank then that oil should be good to use. An even more important factor is if an oil is rated to meet the certification for JASO MA or JASO MA2. That designation indicates that the oil is formulated to work in a bike with a wet clutch. There are some that are rated as JASO MB which is NOT good to use with a wet clutch but is formulated for bikes with a dry clutch. These factors are really the
ONLY factors you need to be concerned about. Use the oil you want within those limits and the engine will last a long time as far as the oil is concerned. The JASO designation is a spec developed in Japan for motorcycles made or designed there and is used in motorcycles from many other countries as well. Most, but unfortunately not all, shops know this about wet clutch motorcycles. There are some riders I know of who have used this wrong kind of oil for their wet clutch engines and have not noticed much, if any, issue. Those are the exceptions and they may yet have a problem in less time than they should. Most who have used this wrong kind of oil end up with an issue with slipping clutches, while in gear, relatively soon after an oil change using those oils. Soon as in only several hundred miles or less.
Here is a graphic that shows what I am talking about with the API Service circle which is usually on the back label of many bottles of oil. Don't pay any attention to the viscosity shown here but only about that bottom outer ring.