Your trigger is a direct pull trigger mechanism. It is very reliable and very safe. And can be worked on.
First let me recommend you buy a new mainspring. 100 year old mainsprings loose their strength. (You can buy a Wolff and cut coils till you have something reasonable. Their extra pound firing pin springs are too stiff in my opinion.) The reason why buy a new mainspring is that in direct pull mechanisms, when the mainspring becomes weak, the sear drags on the cocking piece surface, giving creep. I have fixed any number of triggers by just adding a new mainspring.
That does not always fix it, so the next step is to polish the cocking piece flat. Enfields are cock on closing, sometimes the surface gets a groove, caused by abuse. Remove the groove and polish the surface, and sometimes the problem goes away.
The next is the stoning of the humps. Basically you want to increase the amount of movement the trigger has in the first stage. You want to have it so when the second stage is engaged, that trigger sear is just on the edge of the cocking piece surface. Stone too much and the trigger sear will be off the edge of the cocking piece before the second stage hump is engaged.
Be careful, once you remove too much metal, you will need new parts.