That's another excellent question.
I know time is limited. You may have encountered this in the context of Tanzania, where you've had Burundian refugees present since 1972. They spoke Kiswahili. They had local businesses. De facto, they were integrated, but in the absence of legal status they weren't able to perform the citizen functions of being integrated.
This is an issue of great debate in terms of economic integration. I'd point you to the work of the Refugee Economies project at the University of Oxford. You may have come across it in preparations for your time in Uganda. I would point to the debate in terms of legal integration versus social and cultural integration.
The simple answer to the question is that the measure of integration depends on who you ask and what interests they want to advance in that argument. Since the 1970s, there's been a rich debate on the conditions under which the presence of refugees and refugee assistance programs contributes to the local economy. Part of integration is being able to contribute to the economy, to be able to invest in the local economy, and to perform citizenship rights.
The simple answer is that legal status is the clear benchmark measure of integration—having durable legal status—but beyond that there are the social and cultural questions of integration, and this is debated in Canada as much as it's debated in Kenya.