That they do, tho I am learning this example. When you are buying a gift, it's always "till", as in the above example. When you are performing for an audience, it is "för" (Han sjungar för oss). When you are doing someone a favor, it is "åt" (Kocken lagar mat åt oss). I'm sure there are other examples, but these are the three I have learned so far.
I've found a very interesting article about these prepositions http://www.thelocal.se/blogs/theswedishteacher/2010/10/12/at-till-eller-for/.
It is clear but difficult to use them.
I wonder if "our child" would be a better translation here. I asked my Swedish husband about this, and he thinks it would be natural to say "barnet" in cases where we would say "our child". But he always gives the caveat that he hasn't really spoken Swedish since he was 9.
So my question is: would "barnet" here only refer to some child that has been previously discussed, or could it also mean the speaker's child, without a prior reference to some child?