for a 3-person team, i'd probably start with Jira Service Management before looking at anything fancier. their current cloud free plan covers up to 3 agents, and the portal/workflow setup lets you keep more detailed internal statuses while showing cleaner customer-facing labels on the requester side. that sounds pretty close to what you're after.

Freshservice would be the other one i'd demo. it supports custom ticket statuses too, and you can choose different wording for what agents see versus what requesters see in the portal, which matters once you start adding things like "awaiting hardware" or "waiting on network team."

the bit i'd test in either trial is whether the portal still feels clear after you've built the workflow you want. lots of tools can add statuses. fewer make it easy for an end user to check progress without needing to ping the helpdesk again.