Barring the damage over time changes that will be implemented in the upcoming 3.0 update...
Ignite refers to the elemental status ailment, which causes burning. Per the Path of Exile wiki:
Ignite is an elemental status ailment associated with fire that causes the affected target to take burning damage (fire damage over time).
Burning is more simple - it is the fire damage taken over time:
Burning refers to any fire damage over time effect.
So based on these 2 definitions, it seems that Ignite is one method which can cause burning. This is further supported by the Mechanics section on the Burning page of the wiki, which lists Ignite as the first source of burning damage:
There are several sources of burning, for example:
- Burning ground (Fire Trap, Flame Dash, etc.)
- Righteous Fire
- Searing Bond
- Scorching Ray
So your take on the difference is pretty close: burning is definitely the fire-related damage over time. However I would characterize Ignite as a Status Ailment rather than a hit. So functionally, you would use an attack that has a chance to Ignite - this, if successful, would give the enemy the status ailment of Ignite, causing them to take burning damage.
You can read in greater detail about the mechanics of Ignite, how it relates to burning, and how other factors (like resistance, critical strikes, etc) affect Ignite here.