Bitcoin is actualy three things combined into one:
1. Coins
2. Protocol
3. Payment network
Except that there isn't any such thing as a "coin" in bitcoins. Only transactions on a payment network that transfer control of value by way of a mutually accepted protocol.
Correct use of terms would be, for example, "I have received some bitcoins over Bitcoin" for the case when you received bitcoins using direct access to
payment network (Bitcoin-Qt or MultiBit software)
Or if you want to be technically correct:
"I have discovered an accounting ledger entry in the publicly shared ledger database known as 'the blockchain' which encumbers value with a requirement to provide a digital signature from a private key that I happen to have access to before I can re-encumber any of that value with a similar requirement."
or "I have received some bitcoins over BTC-e exchange" for the case when you received bitcoins without accessing payment network (transfer of coins happened within BTC-e exchange domain, BTC-e user to BTC-e user).
Or perhaps:
"My account over at the BTC-e exchange has been credited with a promise from BTC-e to transfer some value upon a request from me."
Commonly used term for later case is offchain transaction.
True, that is a "commonly used term". I prefer to think of it as an uninsured, unregulated, un-audited "bank account" with a small foreign company.