This suggestion is mainly directed at newcomers to the industry, who may not have the reputation or experience to be able to credibly push back on many points of their employment agreement. Once you're in a seller's market for your level of skill (as I'm sure you are), you don't need this advice anymore and I'd encourage you not to let your employer restrict your moonlight projects. But particularly if it's your first job, you may have to make concessions - for some folks, "if you want a paycheck, sign here" is a reality.

If nothing else, it is very important that you keep the right to move up and out to a new job. That's the point you should focus on. In a realistic full-time first job, you're not going to work for a competitor at the same time, so it is fine to tell them you won't if that helps the two sides reach agreement.

Good point about not taking "this is the standard agreement" for an answer. That's not good enough. Say you understand that it's a boilerplate legal document, and you'd be more than happy to give them a couple days to double-check your proposed change with their boss, counsel, or whoever they need to. Again, be firm while giving them an avenue to save face, and wait for them to call back with a sudden change of heart.