As I have read you can modify and use the Linux kernel if you do not distribute the binary of that modified version. The confusion that I have is that, suppose that if AWS were to provide a kernel modified version of Linux in their Linux VMs, does that count towards the distribution of a modified binary?
The GPL does allow you to distribute modified binaries, but only as long as you also offer the source code. For the GPLv2, this is discussed in section 3: Either you distribute the binary together with the complete source code, or you distribute the binary together with a written offer for the complete source code.
The relationship between GPLv2 licensed software and PaaS providers like Amazon is a bit complicated because the GPLv2 does not contain a definition of the term “distribution”. (In contrast, the GPLv3 contains definitions for “propagating” and “conveying” a covered work.) I think with respect to PaaS, the relevant distinction is who runs the GPL'ed work: the provider or the client.
If the client rents a bare VM and then installs a GPL'ed operating system on the VM, the client must have received a copy of the operating system in order to install it. So the software has been distributed.
If the client rents remote network access to a VM with a pre-installed GPL'ed OS, it is arguably the provider and not the client running the GPL'ed software. In this case, the software wouldn't have been distributed to the client.
For the specific case of Amazon AWS, I do not know the legal structure of the client–provider relationship. However, Amazon's Linux distro available on AWS does seem to provide a command to get the “reference source”. If this reference source is a derivative work of GPL software and is distributed to clients via such a command, then the client would necessarily receive the source under the GPL and would be free to use it according to the license.