They call it transparent because the client does not need to support using a proxy server or even be aware that it is happening.
Transparent proxies are common at corporations that filter web browsing. It is harder to circumvent than DNS blocking.
I suppose that it is no longer a transparent proxy once it starts modifying the requests or responses. But even transparent proxies generally serve an error message in some cases, like when a domain name doesn't exist or a server does not respond on port 80. So they are rarely, if ever, fully transparent.