Is the 3rd Doctor really the 3rd Doctor?

[Brue]

So I watched the entire 21st century series od Doctor Who over the past year, Recently I have been watching the 20th century classic series.
The First Doctor "renews" but the second Doctor is punished - one of his consequences is they change the way he appears. He did not actually regenerate. So is it really a "new" Doctor?

[Mikey]

The Second Doctor went through a forced regeneration.

In less nice words the Time Lords killed him and then he regenerated into the Third Doctor.