Scenario: I'm 20 (pursuing a degree in Computer Engineering) and a self-employed IT consultant with VAT number and everything needed here (Italy).
For the past 2 years, I've been working for BigCompany (very big company), but in reality my invoices are being sent to SmallCompany (very small company), which is the actual provider of consultants for BigCompany. The fact is, I have basically no contact with SmallCompany and have been there just a couple times. As far as I'm concerned, it's like they don't exist. They're just the mean by which I got to work with BigCompany after a successful internship: I talked with BigCompany, they reached an agreement with SmallCompany, I work for BigCompany and send my invoices to SmallCompany.
Now, the question is, am I wrong in writing on my resume that I'm a consultant for BigCompany? To me, it's great because having that big name there makes a really nice impression, but I also know that technically there's a third party in between. But then again, I even have a BigCompany email address and have never been told anything about what's written on my LinkedIn (and I have more than a couple contacts from BigCompany).