Agent terminology can be a bit confusing. Main thing is Azure Monitor Agent (AMA) is new. Log Analytics agent or Microsoft Monitoring Agent (MMA) is the legacy version.
The new agent is the AMA. It is only available as an extension for servers (no stand-alone MSI/EXE installer). It can be deployed directly to Azure VMs and to servers using Azure Arc. This of Arc like WSUS for admin software. It allows Azure to deploy and manage a limited set of admin software and audit policies.
The MMA is the older/legacy agent. Very similar to the SCOM agent. It will be phased out as a cloud-facing agent, August 2024. Customers should start migrating away from MMA early if possible.
There is a special AMA workstations (win10-11). It can be installed as an MSI but only works on AAD joined devices.
The gateway is an optional service to redirect AMA\MMA traffic. Basically, an agent designated to act as a proxy. Both AMA/MMA support this concept/service.