I am wondering whether there are any more experienced testers in your organisation you could talk with? Most of the terms you use mean different things to different people, i.e. a generic lecture through this discussion forum might not solve your problem.
My understanding of "agile" is that testers and developers work in small focus groups close together. It might pay, if you find out what "your" developers expect from you - this might be more useful than seeking generic definitions!
Just a few words to the terms "test strategy" -> "test scenario" -> "test case":
If you want some generic definitions than you might e.g. refer to IEEE Std 1012-1998 (Standard for Software Verification & Validation) - I doubt however that this will help you in any way solving your problems, i.e. I don't recommend that at this stage.
If you look at any testing assignment than there are (for any non-trivial task) an (in practical terms) unlimited number of possible variations of input and timing combinations. That's why you need a test strategy - it is a reflection on which of these unlimited combinations are either most likely to reveal bugs - and/or which of these combinations are critical for your customer (i.e. you really want to make sure that they work).
Your test strategy specifies test scenarios - take them as groups of test cases which have something in common - e.g. looking at a particular feature, or having the same (or similar) input paramenters or whatever you choose to sort them.
These test scenarios consist than out of individual test cases.
You might find that some (many?) organisation use the term "test design" instead of "test scenario" (as used above) - and I am sure that there are plenty of other definitions for "test scenario" as well - i.e. you better find out, what your organisation means with these terms!
Commonly a test scenario is similar to the title or high-level description of a test case. Although one test scenario can have multiple test cases and viceversa. The best way to understand it is that it is the difference between 'what' versus 'how'. The test scenario tells you what you are going to test. Test cases typically contain the detailed steps for how you are going to test. Both are generally mapped to requirements.
The following are some examples of a test scenario:
1) Create a customer log using a negative HD ticket id.
2) Create a customer log using a HD ticket id between 1 and 99,999.
3) Create a customer log using a HD ticket id of 100,000 or greater.
In the test case you would define the detailed steps to perform the test, the test data/values to be used, and the expected results. Good practice is to also capture the result set as evidence of the test.
Personally speaking, I think it is very good practice to write test scenarios. It keeps you from spending your time writing out detailed test cases only to have someone tell you that they are invalid scenarios. "....they don't use the product this way."