No, I'm not wrong. Just 'differently correct'.
"When you choose this domain at first level ..."
I read that to mean "at first character level". If it meant "at first level of cleric" there would have been no need to qualify it. They could have just written "When you choose this domain" and left it at that.
But, as usual, "It's up to your DM." If the DM is me, the answer is "no"; if the DM is you, the answer is evidently "yes" and that's fine by me. But don't tell me I'm flatly wrong, because that's just plain rude.
You may wish to apologize for your dogmatic statement.
You may be "differently correct" about Life, but, as stated,
none of the three other domains which provide Heavy proficiency (War, Tempest, Nature) would be affected by that reading--in fact, they would be further entrenched. Whereas, again as stated, the Trickery and Light domains would lose meaningful but much smaller benefits. There would also be ripple-out effects for a number of other classes: Barbarians (Berserker loses Frenzy; Totem Warrior loses Totem Spirit), Bard (Lore loses the three bonus skills; Valor loses Medium armor, shields, and martial weapons), Druid (Land loses a bonus cantrip; Moon loses Combat Wild Shape), Fighter (Champion loses Improved Critical; Battlemaster loses Combat Superiority), Monk (Open Hand loses Open Hand Technique; Shadow loses Shadow Arts; Four Elements loses Disciple of the Elements), Paladin (all Oaths lose their Channel Divinity features), Rogue (Assassins lose the Disguise Kit and Poisoner's Kit proficiencies), Sorcerer (Wild no longer triggers Wild Surges), and Wizard (all Wizard schools lose the "memorization time/gold costs halved" feature).
Some of these are especially bad--the Battlemaster (you
don't get maneuvers at Fighter 3) and
all Monk traditions (you can't spend ki on spells/Open Hand effects) are completely neutered, while the Wild Sorcerer loses its main "twist." There's also the unnecessary and weird nerfs, like taking the extra 3 skills from people who MC to get Lore Bard. Is it really worthwhile to fix 1 out of 4 problematic Cleric subclasses, when by doing so you make MCing into 19/38 subclasses wonky or even outright broken? Arguing that "starting at level N, when you choose this subclass" (or equivalents) = "only at character level N, not if you multiclass" causes a
lot of problems while failing to even fix the majority of the problem it's addressing.
I think it's quite telling, actually, to compare Life Cleric to Tempest Cleric, or Battlemaster Fighter to Eldritch Knight Fighter. Both types of Cleric give essentially identical "Bonus Proficiency" benefits, except that Life uses the "When you choose this domain at 1st level..." phrasing, while Tempest uses simply the "At 1st level..." phrasing. Also, I only say "essentially" identical because Tempest gives another proficiency (martial weapons) that Life doesn't. Both types of Fighter provide the core feature that makes them
different from other Fighters at Fighter 3, but the Battlemaster's maneuvers/dice/DC/etc. opens with "When you choose this archetype at 3rd level..." while the EK's spells opens with "When you reach 3rd level..."
Given that the majority of the game's subclasses lose something, and some lose a very
important something, I
don't actually consider your reading "differently correct." It carries far too many complications for me, while only "fixing" a single subclass in the bargain. Much better, and simpler, to just say "characters who multiclass into Cleric do not receive bonus Heavy armor proficiency, regardless of their listed Domain features." That keeps the rule isolated to the only class that causes this problem, fixes
all the subclasses rather than only one of four, and remains a simple rule to follow.