Salesforce Governor Limits – Number of Synchronous Concurrent Requests for Long-Running Requests

Salesforce provides a decent explanation of transactions:

What is an Apex Transaction?

An Apex transaction represents a set of operations that are executed as a single unit. All DML operations in a transaction either complete successfully, or if an error occurs in one operation, the entire transaction is rolled back and no data is committed to the database. The boundary of a transaction can be a trigger, a class method, an anonymous block of code, a Visualforce page, or a custom Web service method.

All operations that occur inside the transaction boundary represent a single unit of operations. This also applies for calls that are made from the transaction boundary to external code, such as classes or triggers that get fired as a result of the code running in the transaction boundary. For example, consider the following chain of operations: a custom Apex Web service method causes a trigger to fire, which in turn calls a method in a class. In this case, all changes are committed to the database only after all operations in the transaction finish executing and don’t cause any errors. If an error occurs in any of the intermediate steps, all database changes are rolled back and the transaction isn’t committed.

So, to answer your original question, that will be treated as two transactions. Clicking a button to fire off a single call works as one transaction that will complete. When you click it again, it will be another transaction.

From Salesforce Limits Quick Reference Guide - API Query Cursor Limits

A user can have up to 10 query cursors open at a time. If 10 QueryLocator cursors are open when a client application, logged in as the same user, attempts to open a new one, then the oldest of the 10 cursors is released. If the client application attempts to open the released query cursor, an error results.

So the cursors are a rolling limit where only the 10 most recent are kept for a user.

Also, if the QueryResult.queryLocator is null you haven't consumed one. In my testing when QueryResult.done was true with the first QueryResult there was no queryLocator defined.

As you point out, Concurrent API Request will only be an issue if they take longer than 20 seconds.

You are more likely to run into the rolling 24 hour API limit that depends on how many seats you have. See Total API Request Limits.