Abhdhamma Vibhanga 12, V&V in jhānas
first jhāna gloss V&V, (2nd jhana same gloss)

Wrong English translation by U Thittila 1969

“Accompanied by initial application, accompanied by sustained application” means: There is initial application; there is sustained application.

Therein what is initial application? That which is mentation, thinking, thought, fixation, focussing, application of the mind, right thought. This is called initial application.

Therein what is sustained application? That which is searching, examining, constant examining, scrutinizing, constant connection of (and) constant inspection by consciousness. This is called sustained application. Thus of this initial application and of this sustained application he is possessed, See section 357. furnished. Therefore this is called “accompanied by initial application, accompanied by sustained application”.

Lined up next to Pāli

♦ 565. “savitakkaṃ savicāran”ti	“Accompanied by initial application, accompanied by sustained application” means:
atthi vitakko, atthi vicāro.	There is initial application; there is sustained application.
♦ tattha katamo vitakko?	Therein what is initial application?
yo takko vitakko saṅkappo	That which is mentation, thinking, thought,
appanā byappanā	fixation, focussing,
cetaso abhiniropanā sammāsaṅkappo —	application of the mind, right thought.
ayaṃ vuccati “vitakko”.	This is called initial application.
♦ tattha katamo vicāro?	Therein what is sustained application?
yo cāro vicāro anuvicāro upavicāro	That which is searching, examining, constant examining,
cittassa anusandhanatā anupekkhanatā —	scrutinizing, constant connection of (and) constant inspection by consciousness.
ayaṃ vuccati vicāro.	This is called sustained application.

Show how it's wrong with a similar example:

wrong way to translate 'food': (as dog food pellets)

and what, monks, is dog-food-pellets?
dog-food-pellets is nutrients,
vegetable, fruit, beans,
health-food, junk-food, dog-food-pellets,
human-food, worm-compost, cat-food,
that, monks, is dog-food-pellets.

correct way to translate 'food':

and what, monks, is food?
food is nutrients,
vegetable, fruit, beans,
health-food, junk-food, dog-food-pellets,
human-food, worm-compost, cat-food,
that, monks, is food.

See the logical error?

It's not easy to detect if you're not really looking for it, because the wrong translation has an awkward meaning,  but can still be somewhat coherent. To spot the error, you have to know what the original word being translated was.

Food is not equivalent to dog-food-pellets.
Dog-food-pellets are a subset of food.
So if we translate the word "food" as "dog food pellets", that's extremely wrong.

Similarly, "initial application" is a special additional meaning to the normal meaning of vitakka/thinking defined by Abhidhamma.
"Thinking" is not a subset of "initial application".
"initial application" is a subset of "Thinking".

Similarly for vitakka & vicara (V&V)

vitakka and vicara has established meaning in the EBT, everywhere outside of jhana and inside of jhana, it means "thinking & evaluation".

Vism. is redefining vitakka & vicara to INCLUDE a new special meaning for their redefined first jhana, that only applies within that redefined first jhana. So the correct way to translate the word vitakka, would still be "thinking".  Both Vimutti-magga and Vism. are using that Abhidhamma gloss of vitakka and vicara that U Thitthila mistranslates, so the vimuttimagga would no longer work properly if they went by U Thittthila's mistranslation.

Corrected version of U Thitthila's mistranslation

♦ 565. “sa-vitakkaṃ sa-vicāran”ti	“Accompanied by thinking, accompanied by evaluation” means:
atthi vitakko, atthi vicāro.	There is thinking; there is evaluation.
♦ tattha katamo vitakko?	Therein what is thinking?
yo takko vitakko saṅkappo	That which is mentation, thinking, resolves,
appanā byappanā	fixation, focussing,
cetaso abhiniropanā sammā-saṅkappo —	application of the mind, right resolve.
ayaṃ vuccati “vitakko”.	This is called thinking.
♦ tattha katamo vicāro?	Therein what is evaluation?
yo cāro vicāro anu-vicāro upa-vicāro	That which is searching, evaluation, constant-evaluation, evaluative-scrutinizing,
cittassa anusandhanatā anupekkhanatā —	constant connection of (and) constant inspection by consciousness.
ayaṃ vuccati vicāro.	This is called evaluation.

The highlighted words correspond to Utitthila's mistranslation of V&V as "initial application" and "sustained application".

Why is it deliberately mistranslated?

Because vitakka in the EBT means 'thinking' in every single case, whether inside or outside of jhana. So U Thitthila is deliberately mistranslating it to indoctrinate people with the Vism. redefinition of first jhana.

If you compare to Vimutti-magga for example,  composed a few hundred years earlier than Vism. and also based on Abhidhamma, there you can see that of the 40 meditation subjects, some subjects such as metta bhavana, 4 brahmavihara, uses vitakka = 'thinking' as a way into first jhāna, while also supporting the 10 kasina meditations which use the specialized  meaning of vitakka as "initial application" only in that special case.  In Vism., they change the metta practice and 4 brahmavihara  meditation to disallow the thinking, and you now have to use kasinas visualizing the beings you're directing to in order to be in 'first jhana'.

Conclusion

So the reason for the mistranslation of vitakka & vicara, is you're being groomed. You were used to eating human food from EBT (early buddhist texts). Late Theravada Abhidhamma is grooming you to only eat dog food pellets, and redefine the meaning of 'food' as 'dog food pellets.' You will eat dog food, and you will like it.