As we know, Half-Life was made with really heavy modifications of the Quake Engine, with some codes from Quake 2 and or QuakeWorld. However, there are things that set Half-Life Engine (or alternatively, GoldSrc) apart from normal Quake Engine. One immediately noticeable change is the characters non wobbly animation thanks to in-house build skeletal animation, support for 16-bit coloring (in initial Half-Life Alpha documentation, later versions of retail HL are capable of doing 32-bit coloring within OpenGL) and the presence of Direct3D renderer (in early versions, after SteamPipe update the D3D renderer was removed due to the complications of modern hardware using DX6/7 mode).

Back in the days of pre-Quake era, FPS games only used simple side sway to make weapon seem "moving while running". These weapons, are just BMP(?) files of weapons' upper receiver being tossed from left to right and back. In a world where true 3D FPS games weren't exist, this is a sufficient move. However, with the introduction of Quake in mid 1996, those swaying movements are not enough to make players believe that they're actually carrying 3D weapon models. So, Id Software decided to make the weapon sway (now called viewbob since it bobs) forward and backwards instead of side sway, and adding viewroll when they strafe to right or left.

Some time after the release of Quake, the newly founded Valve Software wanted to create a whole new innovation in FPS games, mostly in realistic (for the plot setting given and technology at the time) guns and seamless story. Valve, knewing that default Quake's bobbing isnt enough, decided to re-code the animation to make it more realistic. Initially, Valve not only modifying the bobbing cycle and time, but also change the direction from forward and backwards to tilting to side. However, after some later patch within Half-Life WON version and Steam, the direction is returned to Quake's forward and backwards.

Temporary Effect (defaults after you quit the game):

First, check up your cl_bob and cl_bobcycle values by simply typing them to the console (note for WON users you have to add -console parameter into Half-Life's shorcut first).
Second, re type these values into each of those cvars: 0.02 for cl_bob, and 0.6 for cl_bobcycle. These are default values for Quake's bobbing.


Semi-Permanent Effect (defaults if you removed them):

WON: Right-click on the Half-Life shortcut, then go to Properties. in the box for "Target", after hl.exe, add these parameters: +cl_bob 0.02 and +cl_bobcycle 0.6.

Steam: Open Properties by clicking the gear button (assuming that you're into the library page for the game, otherwise right click it), in the launch options, add the parameters from above.

IMPORTANT: In case your values do not default, then type these values, 0.01 and 0.8 for cl_bob and cl_bobcycle respectively.

Console commands:
https://quake.fandom.com/wiki/Console_Commands_(Q1)