In case anybody's interested and/or has feedback/corrections:
I've found that lose-track involves the following specific categories of cards. The blue ones can actually lose track.
1. THRONE ROOMS: Abilities that play a card twice (Throne Room etc.)
2. Some triggered abilities triggering on when-gain, when-discard or when-trash
2a. WATCHTOWERS: when-gain abilities that move the gained card (Watchtower, Innovation etc.) [Villa only tracks itself]
2b. INNS: when-gain abilities that may happen to move "themselves" (Inn, Blessed Village, Cursed Village)
2c. SHEEPDOGS: when-gain abilities that may happen to move the gained card (Sheepdog; when gained on when-gain: Cavalry, Inn, Blessed Village, Cursed Village; with Way of the Mouse: Black Cat, Falconer, Sheepdog)
2d. NOMAD CAMPS: when-gain abilities that put a card on top of the gained card on your deck (Nomad Camp, Rocks, Cursed Village, Blessed Village, Mandarin)
2e. BORDER VILLAGES: when-gain abilities that gain a card on top of the gained card in your discard pile (Border Village, Death Cart, etc.)
2f. SCHEMES: when-discard abilities that move the discarded card (Scheme, Hermit etc.) [track only themselves, except Scheme, Prince and Way of the Frog]
2g. FAITHFUL HOUNDS: when-discard abilities that move the discarded card (or may happen to move it) outside Clean-up (Faithful Hound, Village Green) [Both only track themselves]
2h. TUNNELS: when-discard abilities that gain a card on top of the discarded card in your discard pile (Tunnel)
2i. FORTRESSES: when-trash abilities that move the trashed card (Fortress, Possession) [Fortress only tracks itself]
3. NECROMANCERS: Abilities that play a card without putting it in play (Necromancer, new BoM, etc.)
4. Abilities that move a card twice
4a. PROCESSIONS: abilities that play and move a card (Procession: trashes, Counterfeit: trashes, Prince: sets aside)
4b. REPLACES: abilities that gain and move a card (Reap, Replace, Summon)
4c. VASSALS: abilities that discard and move a card (Vassal, Warrior)
4d. THIEFS: abilities that trash and move a card (Thief, Noble Brigand)
5. TURTLES: Abilities that move the played card. (Way of the Butterfly, Way of the Horse, Way of the Turtle)
And these interactions can lead to losing track:
1. A THRONE ROOM plays a card which is moved from play: the THRONE ROOM loses track. The card's ability, and all TURTLES, also lose track when the THRONE ROOM plays it again.
2. A WATCHTOWER moves gained card: all other WATCHTOWERS lose track.
3. A SCHEME moves discarded card: all other SCHEMES lose track.
4. A FAITHFUL HOUND moves discarded card, all other FAITHFUL HOUNDS lose track.
5. A FORTRESS moves trashed card: all other FORTRESSES lose track.
6. An INN is gained and moves itself, or a SHEEPDOG moves the gained card: all WATCHTOWERS lose track.
7. A card is gained onto your deck, and a NOMAD CAMP causes it to be covered: all WATCHTOWERS lose track.
8. A BORDER VILLAGE is gained onto your deck, gaining another card, and a WATCHTOWER moves the last card onto your deck: all WATCHTOWERS lose track of the BORDER VILLAGE.
9. A NECROMANCER plays a card: the card's ability, and all TURTLES, lose track.
10. A PROCESSION plays a card which is moved from play: the PROCESSION loses track.
11. A REPLACE gains a card, and a WATCHTOWER moves it: the REPLACE loses track.
12. A REPLACE gains an INN which moves itself, or a card that is moved by a SHEEPDOG: the REPLACE loses track.
13. A VASSAL discards a card, a FAITHFUL HOUND moves it: the VASSAL loses track. The card's ability, and all TURTLES, also lose track when the VASSAL [only Vassal] plays the card.
Obsolete since the 2019 errata: A card is gained to your discard pile, and a BORDER VILLAGE causes it to be covered: all WATCHTOWERS lose track, and all REPLACES lose track.
Obsolete since the 2019 errata: A VASSAL discards a card, a TUNNEL causes it to be covered: the VASSAL loses track [only Warrior]
(With Way of the Mouse it's possible to make THIEFS lose track by playing a card that gains the trashed card.)
EDIT 2020-01-21: Added interaction #8. Added to "2b INNS" and interaction #6.
EDIT 2020-03-04: Split "2b INNS" into "2b INNS" and "2c SHEEPDOGS". Updated interactions #6 and #12.
EDIT 2020-07-13: Updates for Menagerie