Is it normal that my baby behaves better for other people than me?

Yes. Sure, we know what we can get away with our parents, but we don't know what others will let us do. Our own kids push our buttons more, and we can be real suckers sometimes (especially me with my own girls). It's a 2 way street. We are probably lenient with our own kids, and our kids push us to the limit as well. But, it's normal and don't worry about it.
Yes. When our infants cry we run to them and tend all their needs, transport them, nurture them & treat them like little kings/queens. They know how to push our buttons and expect we will do what they want...Because that's what we taught them. When they start to walk & talk we are pissed that they won't mind!? They don't expect others to reliably tend to their whims & seldom push it.