The "trivially true", but probably unsatisfying answer, is simply "because the relevant parties agreed that to be the case".
The rest are just narratives that the relevant parties agree or don't agree to use as an excuse.
The thing is "international law" behaves a lot different from what an individual might perceive "the law" to be. For example from the level of an individual, there is one or several super structures that: make law, enforce law and pass judgement.
When it comes to international relations there is no such thing. There is no (official) world government, no world police and no world court. There is something of a mutual anarchy (in the sense that there is no ruler). Sure countries can agree to have laws, courts and declare things crimes; and all of that is happening. But there's no obligation for anyone to accept any of the laws being passed, nor is there any dedicated institution that is able to penalize violating the law. Like with the laws and agreements those judgements might be passed but it's upon the perpetrator to accept them.
So what this boils down to is a complex web of reputation, agreements and powers.
Sure a perpetrator could technically say "laws don't apply to me", but practically that might mean that they are seen as unreliable and cut off from international trade and beneficial agreements or that other countries decide to go to war with them as they violate their interests and that further different countries will see that as a "just war" and supply them or remain neutral. So while most of these agreements are just as binding as you let agree to them being binding, they nevertheless posses some binding nature due to the fact that it would severely damage your reputation and international standing if you'd violate them and the more interconnected things get the more unlikely it seems to be for one country to opt out and just flip off every other country.
Now with respect to what constitutes an act of war. Same scenario, of course there are rules and regulations for what are good and not-so-good reasons to declare a "just war". But in effect a war is declared when a war is declared or when the first red line is crossed and that can be whenever one of the relevant parties says it is.
Objectively whether arms dealing counts as participation in war or not solely depends on the perspective. Like the arms dealer would obviously like to treat it as "just a job or just business as usual" and not get involved in all the nasty responsibilities and things that might hurt the bottom line. While if you are at the end that benefits from those deals you might agree with that neutrality not necessarily because you like it or actually agree with it, but because it's still better than not doing so, in the end it's still beneficial to you. While if you are on the receiving end of those arms you might argue more from the perspective of the arms dealer being part of the extended military supply line and declare it a legitimate military goal to cut the enemy off from that. And that's technically not wrong.
Practically though, unless you want to go for global tyranny, you still need to find agreement with your perspective in the camp of the other states that are currently not involved in the war, so it would make sense to frame your perspective as a universal doctrine. And in that case Russia wouldn't have a great case for itself, given that during the cold war there were lots of proxy wars where "just supplying one side with arms" was done by both sides and therefore explicitly declared NOT to be an active party in the war. Obviously that's a blatant lie, but if the relevant actors in that scenario agree on the lie, then it's de facto true for the sake of argument.
Because similarly to the cold war, the question isn't whether you COULD declare that an act of war (you always can whether that makes sense or not), but whether you actually WANT that.
And while Russia probably doesn't like other countries supplying Ukraine with weaponry, fighting an open war against all of those countries, at once, would mean a whole lot more weapons and soldiers on the battlefield. So the effect of that would be even worse. So it's a narrative they can use in their domestic propaganda, but internationally it probably only serves as a "red line". In the sense of "if you breach this line, we will attack you". Though declaring a red line in hindsight only works if you already want to attack and look for an excuse so if that's not the goal, you can only apply that to interventions that have no yet happened.
That being said there's still an intrinsic difference between supplying arms and supplying combatants and that is that the latter are human being and more specifically citizens and thus representatives of your country. So an attack on them either means an attack on your country, which technically either makes them moving "no shoot zones" or increases the threat of a necessity for war so massively that you might already call the deployment an act of war, or you'd have to basically disavow your citizens and the protection of your citizens from hostile foreign interference.
So that's no longer between you and the respective enemy but between your country and it's citizens and as someone else noted for every battalion there's several battalions of relatives.
So while supplying arms and sending troops are probably both an involvement in combat, at least the former has an agreed upon plausible deniability while for the latter such a claim would become completely absurd (which is not to say that countries aren't willing to walk on these egg shells if they think they could do it). Like how Russia apparently send unmarked soldiers and disavowed relations to them in order to pretend an invasion is not an invasion. Though that has nothing to do with law, reason, logic or any of that sort and purely with the question what countries think they can get away with and with what other countries let them get away with.
