Regular solder - 60%lead 40% tin. 450 deg F, - low strength.
"Silver solder" 95%tin, 5% silver (% varies) - 2-3 times as strong as regular solder. Needs 450-600 deg F. and more aggressive flux. Great for restoring old "regular solder" joints on shotgun ribs, etc.. This is HI-Force 44 and Stay Silv type solder.
"Silver braze" alloy 15-45% silver, remainder copper. Considerably stronger than "silver solder". The term is not properly used and it is often called "silver soldering" in the vernacular. Requires borax based flux and 1200 deg F. or so, depending on percentage of silver in the alloy. More silver in the alloy makes the solder able to flow into small spaces, less means thicker solidus state and gap bridging ability.
Copper or brass brazing - brass (copper and zinc) or copper alloys - 1850 + deg F. to flow and strong borax flux or inert atmosphere (furnace braze usually in the gun trade). Somewhat stronger than Silver braze if properly done. Could allow some carefull heat treatment after joining. 4140 steel in the gun trade is hardened below 1500 deg F. and is tempered around 900 deg. F. so it should survive normal heat treat. I'm working from memory here, but I believe the figures are approximately correct.
TIG welding - tunsten inert gas - 2500 + deg F., and actual melting of steel with or without filler metal, shielded by argon or other inert gas mixture. Fully as strong as the base materials (with some heat affected areas).
All in all, the bolt handle in the common shop should be put back on using silver braze. Silver braze pastes will work if things are dead nuts clean and things are fixtured and already "tinned". I haven't had much luck with pastes when used between raw steel surfaces, but it works fine on raw brass and copper and for repair use of tinned materials. My prep is usually to clean with solvent, then abrade, as sometimes the solvents will degrade the surface if you use them after the abrasion. I prefer to "capillary" a high silver alloy into the joint, but there are many good methods (even when I use a solder paste I usually also make sure the joint is full by addding a little extra solder to ensure a small fillet). Both heat control and proper fixturing are important.
If you have the time and money the TIG weld attachment is by far stronger and better, but this needs to be done by someone who has good equipment and solid experience.
Scott