No, you'd be British. People that claim to be English when neither parent is English is plain wrong.
Rubbish. A person is not their parents.
Why do you make that distinction between British and English? If a person whose parents aren't British can be British, why can't a person whose parents aren't English be English?
Why do you define "English" by who a person has sex with? That is what you're doing, as you define it by parenthood.
Scenario 1: Some people emigrate to England from what was a British colony prior to the 1950s. They grew up in an area with a strong British and particularly English influence, speaking English as their native language and with many English customs. On emigrating to England, they become entirely English. Cricket, tea and scones, the whole nine yards. Many of them have children together after coming here. Some of those children grow up and have children of their own. In some cases, both parents were children of people who came here in the 50s as described above. So by your definition, these very English people born in England to people born in England are not English.
Scenario 2: Someone emigrates to England but has no connection to it at all. They don't identify themself as English. They speak English as a foreign language and only when necessary. They live in an enclave that is identifiable as part of England only by the weather and most of the architecture. Different language, different clothing, different customs, different everything. Someone English converts to suit them and they have a child together. That child has no connection to England - they don't identify themself as English, they speak English as a foreign language and only when necessary, they grew up and live in an enclave, etc. By your definition, they're English.
Your definition does not match reality.
