Bao wrote:"Grasp the sparrows tail" is obviously just a misunderstanding of, or maybe originally a joke on, "lazily tying the coat".
From Lan Zha Yi, 懒扎衣 (lazily tying the coat) to Lǎn què wěi (揽雀尾). In Chinese, the character 尾 or wěi (tail) can also be pronounced as "yǐ" as in "yǐ ba" which means also means tail. So you can see that someone which another dialect could easily misunderstand the name.
Is it obvious?
Yang Cheng Fu says the sparrow's tail is the opponent's forearm.
Like the other animal encoding it refers to the technique, he says, but he is also a proven liar.
Lan Zha Yi, as I know it from the Hao I'm just beginning to study, is equally as important in Hao style as GST is in Yang, but it is so much different. In the Hao it is a much simpler set of movements that have more in common with fair lady works the shuttles and focused more directly on internal technique (raising, connecting, opening, closing) than the tactical considerations of push hands (peng, lu, ji, an and stick, adhere, join, follow) you find in Grasp Sparrow's Tail.
The Wu family was learned and literate, to dismiss the change between the two as a misunderstanding is a bit of a stretch.
However... you look at like the Yang Jwing Ming lineage and his first movement after commencement is called Grasp Sparrow's Tail right and Left BEFORE the peng lu ji an sequence. The part that looks kind of like part wild horses mane, sometimes like diagonal flying, but different subtlety. His is MUCH more exaggerated than how I learned it. I learned it as ward off left in the Yang long form.
You can do THAT movement like the Hao Lan Zha Yi and it makes perfect sense internally and externally. More so than the current accepted normal idea of what's happening there, IMHO and it makes even MORE sense than the confused parting mane/diagonal flying/ward off that we usually consider it to be.
As far as "embrace tiger to return it to the mountain" the tiger is the opponent and the mountain is the earth. Your left hand is low to add leverage to the lower back or leg, right leg behind them, right hand across the upper body. Trip with the foot, push the right hand out and down to your hip, lift the left arm up to shoulder height. They better hope they have a good chiropractor.