Both of the approaches you described are valid.
For small web APIs that include their own application server (e.g. Spark, Javalin, Spring Boot) on Linux servers, I start a screen session and run java -jar. This allows me to exit the SSH session (disconnect from the server) without terminating the program.
For other Spring applications that don't include an application server, I package the code into a WAR and copy it to an application server's deployment directory. For Tomcat, that's webapps. The application server can then read the WAR and spin up a running instance (assuming hot-deploy is enabled).
Tomcat in particular also has a web page where you can upload your WAR file to deploy it.