The release of Half-Life 2 (Source Engine), DOOM 3 (id Tech 4), Far Cry (CryEngine) along with UT2003'/'4 (Unreal Engine 2) will always be the marking line in the sand for me that determined the arrival of what I consider to be "modern" gaming proper (it's a broad term in general in conversations, but WHEN did "modern gaming" start? That's when it did, for me). So indeed, not only were the FPSes released that year actually good and impressive in their own ways, but their respective engines would forever change the future of video gaming.
The Source engine established 'proper' physics (to the point of interactivity that it reached with the environment, with physics-based gameplay elements along with physics-based puzzle solving; it was revolutionary), id Tech 4 improved on overall lightning and shadows effects like never before seen, CryEngine for FarCry 1 showed us all some very impressive draw distance rendering and would lead to Crisis no longer than 3 years later while being a solid contender next to Half-Life 2 and DOOM 3 at the same time and all that from a team that basically no one had heard of before; and finally Unreal Engine 2 would of course lead to UE3, which to this day remains one of the most utilized games engine in video gaming history (in large part thanks to how much it was largely used during the XBOX 360 / PS3 / Wii life spans, and also explains why so many games - particularly action games - during that generation all looked so similar and 'Gears of Wars'sy').
Anything prior to the arrival of those game engines makes games feel, move and look so differently. It was akin to the shift between semi-3D (that we simply did consider 3D anyway) on PS1 and Saturn, and 'actual' 3D when the N64 arrived. Or another good example would be to compare what we had on N64 to - and until the arrival of - the PlayStation 2 (or Dreamcast if you prefer, it's a similar comparison). That sort of major shift is what I'm referring to. That's what happened in 2004, in comparison to what we had prior. I think only UT2003 was earlier enough (coming out prior to those 2004 games) and was very impressive when it came out, just as a small heads up for what would be coming just over one year later (with FarCry in March 2004), since it was already using Unreal Engine 2 and that was as early as September 2002, a little ahead of the curve so to speak.