There is nothing inherently wrong with what the service manager told you. (As an aside, service managers and mechanics are looking for add-on work. They get paid more money for this work than work you bring to them to do. If he says it's not a problem, believe them, because if they could make some money off of you, they'll do it.)
If it serves your peace of mind to add some coolant to the overflow, there is no issue with this. Follow the vehicle owners manual as to when to check it (hot or cold) and add fluid accordingly. Anywhere between the marks is golden. Check it every so often to ensure you have the right amount of fluid in it. If there is an issue, it will all of a sudden be too high (over flowing high) or too low (bone dry). If it does this, then worry about something being wrong.
You can add distilled water (which you should use instead of tap water ... tap water will have minerals in it which can react with metals in your engine and radiator and cause sediment in your coolant system). I think a better choice is to purchase 50/50 pre-mix antifreeze/distilled water. Purchasing a gallon of it, topping off your system, then leaving the rest for a later date when you might need to add a little more. My understanding is there is no shelf life on antifreeze, so put the top back on it and be good. There are three or four different types of antifreeze you can put into cars these days, so ensure you are using the correct antifreeze for your application. These are all color coded, so it should be an issue (green, orange, yellow, and possibly red). Each one is specific to certain vehicles and you don't want to mix them. It's not that you'll cause the engine to blow up, it's just that the coolant will not behave like it is supposed to for longevity and cooling effect if they are mixed.