So according to the documentation for the SOAP api for Salesforce, if I log in and start a process that takes 1 minute before I log out, then someone else logs in with the same credentials and begins another process that takes 30 seconds before logout, when the second user logs out, it will kick out the first user as well.
Does the REST api have this issue? Is there a way around it?
This is really a session management issue more than any specific API you want to call using that session. that said, you have 2 easy options.
a) don’t call logout, just let the sessions expire based on inactivity, that way if the session happens to be shared by 2 different integrations it won’t cause a problem.
b) switch to OAuth to do your authentication, and give each integration its own OAuth client/remote access entry. (sessions are not shared between different oauth clients).