With Winchester it was somewhere around 1992 for the cross-bolt safety and around 2003 they switched to a tang mounted safety. Marlin started earlier - early 80's. Although somebody else will have to say whether the Marlin is a trigger lock safety or a hammer block safety. I haven't shot one of the Marlin lever actions since the mid 70's.
The Marlin 336 family (that includes the 444s and 1895s) got the cross-bolt safety in 1983 (serial numbers with first two digits "17" or less). Some rifles, primarily 444s and 1895s, were built on old receivers and sold without safeties into 1984, but the first year for the lawyer button was 1983.
It is a hammer block safety. You can still pull the trigger and drop the hammer. It just smacks the safety instead of the firing pin - if operating correctly - while your deer bounds off into the woods, because you forgot that you put the stupid safety on in the first place.
I, personally, have collected a few "brush guns" over the last 5-6 years.
The first one was a Mosin M38. It works, but after carrying a Marlin 336 on an Elk hunt, I no longer cared for the M38's ergonomics. A few things served "brush gun" duty while I figured out what I really wanted.
Now, the "primary" brush guns (mainly for Elk) are an H&R Handi-Rifle in .444 Marlin, an AR-15 in .458 SOCOM, and an under-construction Marlin 336 (with a safety
) that's being converted into a custom model 444SS*. There are also two possible bolt-gun conversions on the table - a Mauser action that could go 'stubby' (18" bbl) .270 Winchester or .35 Whelen, or a Marlin X7 that could be used for a .35 Whelen build.
*(Cornbush has a nearly-complete 336-to-444 conversion, that he did the final fitting, [temporary] finishing, and troubleshooting work on - it developed some issues after initial testing, but we got them worked out.
)