Whenever I do a new weekly attempt for Invincible (271/100) and Life Binder's Handmaiden (260/100), the stat counter does not update with the newest attempt and chat box line also doesn't have note to say for example "Invincible: 271 attempts". I have had to manually update these now for the past few weeks. They are being tracked currently and all the settings seem to be fine, so I'm not sure why just these 2 recently refuse to automatically count and update.

Those items should be tracked automatically using Blizzard's kill statistics. The first step in troubleshooting any issues with them is to verify the statistics are actually updated when you defeat the boss, as Blizzard is known to break them and Rarity will not track attempts in this event :(

You can simply check ingame, in the Statistics tab (next to Achievements) in the Achievement UI.

Okay some more information about this behavior: I tried to kill LK on an alt tonight that only had 3 kills and was being included in the count previously. Rarity and achievement statistics were updated with my alt killing LK, but it appears my main is the only one not being updated/bugged for some reason

Another update: I tried Deathwing on my alt that had never done it before and the count went up in rarity/achievement ui also, so the issue for these particular encounters are definitely linked to my main character it appears.

So upon looking in the game today at my statistics, Lich King 25H kills show 247 (which is where count stopped updating) and 25H Deathwing shows 227 (also where it got stuck) both got stuck earlier this year then like maybe December/January shortly after SL release. Is there a reason the in-game statistics breaks like this? :( Also thank you so much for a helpful response because this issue has been irking me :P

Yeah, that is exactly my point: Rarity takes the statistics from Blizzard, and if they don't update it won't detect the attempt either.

I've no idea why they break the statistics every once in a while, presumably a lot of spaghetti code on Blizzard's end? Usually they stop working when they introduce Timewalking for a raid, but that's no guarantee it won't break in other scenarios, as well.

If only one character is affected I'd first make sure the difficulty is set correctly, ingame statistics aren't increasing either, and once you're sure it's actually broken on Blizzard's end report it as a bug (ingame, Help section of the Escape menu).

Then, all you have to do is wait a few years, or decades, and maybe they will get around to fixing it eventually :S