Should an educator define terms rigorously?
You have two complementary questions here. What should the student do, and what should the professor do. I'll handle them separately. 

Whenever you can't understand what is being asked of you, ask for a better or more complete explanation. You can't fairly answer a question that you don't understand. Of course, you should be aware of standard terminology and should be able to depend on that being a shared knowledge. But even if that is not the case, you should ask, or consult other sources, so that you don't have gaps in your education. 

The question of what the professor should do, however, is a bit more flexible. Mathematicians and other technical people are rigorous about some things because they are more important and generally useful than other things. For example, we are rigorous about the definition of the derivative in calculus and about what it means to sort numerically. But some things don't need the same care. If a professor needs to put a name to something for a single conversation and it isn't a generally useful or well known concept then there may be no need for complete rigor. A shared, but informal, understanding may be enough for the current purpose. 

So, for example, if the professor wants to define something rigorously, I suspect that s/he will want to remember that definition for the long term. If it comes up in an exercise but you will be expected to have learned it well for a later exam, then some rigour is called for. But for the actual example you give, connected sequence is probably enough, with a few examples to get you on the right track. 

However, If you are in doubt about the importance of a term, then ask about that also. "Do we need a rigorous definition here? Will we return to this concept?" 

Sometimes a teaching analogy or metaphor is enough. If everything were made strictly rigorous your education would become much more difficult (more to remember) and for most learners, much more boring. People are good about learning from metaphor. But professors need to be careful not to load too much onto any given metaphor. 

I'll note that rigor in mathematics is mostly a feature of the previous century. People formalized what was, up to then, considered well known without formalisms. Even today, people discovering new mathematics may struggle for a while about what are the correct formalisms for a new concept. It may take a while for it to "shake out". The important things then get properly defined and names that evoke the correct mind set are applied - or we hope so, in any case.