The GPL does not require you to release your modified version, or any part of it. You are free to make modifications and use them privately, without ever releasing them. This applies to organizations (including companies), too; an organization can make a modified version and use it internally without ever releasing it outside the organization.

But if you release the modified version to the public in some way, the GPL requires you to make the modified source code available to the program's users, under the GPL.

Thus, the GPL gives permission to release the modified program in certain ways, and not in other ways; but the decision of whether to release it is up to you.

Exactly.

Though in the "a company or other organisation" situation IIRC your internal users count in the same way so far as the GPL is concerned so you should make the code available to them on request if you don't already and they are free to redistribute under the terms of the GPL. Another important caveat is the "upon request" part: you don't have to release the source to anyone unless explicitly requested so you don't have to release it automatically upon releasing a binary.

Of course if you just use the results for your own needs/entertainment there is no requirement to release anything at all.