On VM/370, the go to text editor was XEDIT. On TSO with ISPF, it built in to ISPF. Both were very similar and you used a 3270 terminal.
XEDIT was rather late in the day, and didn't appear until 1980. Before that there had been EDGAR but that was also a product. There used to be a video showing this, but I can't find it. Perhaps MOSHIX has removed it. Both VM/370 and TSO also had simple EDITORs out of the box....
However that is VM/370 and you asked about OS/360. I think its worth noting that the 3270 terminal and the TSO option for OS/MVT both didn't appear until 1971. So for most sites on-line editing wasn't available until that date. There were terminals, IBM 2741 Golfball typewriters and 2250 screens, but I don't believe these were used for editing. Its worth noting that both APL and CICS were available before TSO and these screens could have been used for these products..
So what was used for editing? Well on both OS/360 and VM/370 we had something similar to the UNIX patch program. On OS/360 it was called IEBUPDTE and on VM simply UPDATE. So your source could be stored on-line, but when you wanted to update it you created an IEBUPDTE task, usually on cards, that would update the source and compile the updated source to a test object library. You could then test the new program. As you found faults you simply modified your update deck, always using the current production source as input. Than when you and the users were satisfied the modifications were correct you could create a new master source, ready to repeat the process.
If this sounds long-winded it was, but computers were expensive, and in the commercial world work was usually tightly scheduled. Where I worked were only one or two slots per week for development and testing so it was a slow process...
.. oh and perhaps I should mention that you could "edit" in a card punch. The 029's keyboard punches that most sites had could copy column by column. You could engineer an insertion or deletion by firmly holding either the old card under the read head, or the new card under the punch head while typing a new character or hitting the space bar.
If you were using a manual punch, and we still had those in 1976/77 when I started being paid to work with computers, one might resort to taking a bit of chad from the box and pressing it firmly into the hole in the card to patch the card. This was usually ok for a few test runs, but not recommended for production....
using a manual punch
using an 029
... but also look at some of the other 029 punch videos....