There's so many terms out there to describe hair color. A lot of them are interchangable and cause a ton of confusion during a consultation. It's best to bring a picture because it's often very hard to verbalize color.
Ombre VS Balayage
Ombre has been around for quite a while now. This term simply means "fading from dark to light". Balayage is a technique. It means free handing highlights. Which means you could use balayage to create an ombre.
Foiling is when a section of the hair is weaved through with a comb and placed in a foil to create dimension. You can create balayage looks using foiling. Balayage is not the ultimate go to for a "natural lived in look" This techniques can be used differently to creates different looks depending on your hair and your hair goal.
Sombre just means "soft ombre" a more subtle gradation of dark to light. Less contrast.
Babylights are tiny tiny blended highlights created by foiling to mimic children's sun kissed natural color.
Tipping is created by back combing the hair and just placing the lightening on the ends of the hair.
Hope this helps with any confusion! Don't worry so much about the terminology because the word you think you want may not be the look you are going for at all! Look for pictures rather than the buzz words to communicate with your stylist.