It's definitely not a 1960, all you can really do is write down the VIN number on the frame and either put it in a post/reply on here or send it to me in a private message and I will send it to a guy I know that knows about 320 VIN numbers and he can likely tell you the exact year it is.
It is obviously missing the VIN tag, as I can see the 4 holes for it on the firewall in one of your photos, these trucks didn't have a tag on the door pillars like later models have, at least my 62/63/64 titled trucks don't, I don't recall about the 65 model year, I have a 62 L320 titled as a 63 as that was the year it was sold.
My 1962 L320 titled as a 1963 has the air filter housing cut off the same way as yours has been cut off, just at a different angle, that is very strange.
Do you have a heater/heater box in your truck?
I don't know the year of truck you have, but it has a spin on oil filter, so it is a newer engine, what engine number do you have, it would be above the distributor on a flat spot just below the head, the left flat spot will say "E1", or "J13" and the right flat spot would be the engine number, it also has six screws holding the valve cover on, you can see the nut on the top middle of my valve cover in front of the oil cap in the photos above, mine has an early engine.
Here is a photo of my engine compartment with the VIN tag where it is supposed to be.