First, you are right, the term sidecar container has now became a word for describing an extra container in your pod. Originally(?) it was a specific multi-container design pattern.
Multi-container design patterns
Sidecar pattern
An extra container in your pod to enhance or extend the functionality of the main container.
Ambassador pattern
A container that proxy the network connection to the main container.
Adapter pattern
A container that transform output of the main container.
This is taken from the original article from 2015: Patterns for Composite Containers
Summary
Your note on
But, According to Istio -Installing the Sidecar, They introduce proxy as a sidecar pattern.
In the patterns above, both Ambassador and Adapter must in fact proxy the network connection, but do it with different purpose. With Istio, this is done e.g. to terminate mTLS connection, collect metrics and more to enhance your main container. So it actually is a sidecar pattern but confusingly, as you correctly pointed out, all pattern proxy the connection – but for different purposes.