A gateway is a device that can send traffic to a different network. The gateway operates on the same data-link LAN as the source device, which sends traffic destined to a different LAN to the gateway.
The OSI model doesn't have a "dialogue layer." I presume the quote means the application layer.
Typically, a proxy server is for an application (usually web browsing, but not limited to that). You would have your web browser (or other application) configured to use the proxy server instead of the default gateway for the host, and the application traffic is then sent to the proxy server that stands in (proxies) for the host when communicating to a web server. The proxy server can then be configured to perform security functions such as denying unsafe web sites, data inspection for malware, etc.
I'm not sure of the source of you quote, but it seems to use its own, non-standard terminology, so I would look for a different learning source.