Interface button to trigger automation- date modified?

Kind of!  You'd need to create a Date field to contain that information, as well as a formula field with "NOW()", and you'd add a step to the automation to update that Date field with the value from the formula field

Showing the date the automation ran's trickier though.  I think I would handle that by having a new table with a single record in it called "History" or some such that would contain that Date and formula field I mentioned above

The Interface would then have a record picker element that would have that record picked, which would allow us to display the "Date" field from that record, showing the date the automation was last run, does that make sense?

@Pam_Siebert 

You could always just set the automation to not run again if it was already run once in the past. You can do this by simply setting a checkbox field or a date field or any other field.

You could always have a field that has two choices "needs contract generated" and "contract generated".  At the end of your automation set the field to "contract generated".  

OR

You could use a 3rd value "contract generation in process".  Create a field that looks for the last modified time of that state field.  In your interface, you can show the state and the time so that your user will know when the state changed.  This is handy if your contract process isn't quick.  If for example it isn't successful, the "time entered state" might reflect an hour.  If it shows a longer than expected time, your user might want to start the process again by clicking on the generate button.  If the contract was successfully created 2 days ago but needs to be regenerated, the user knows that and can regenerate it.