There is value in clarifying what is meant be a specific usage of 'subsidy', (which the linked paper does in the first sentence of its summary). There is value to calling people out for not doing this clarification when they use the word in its less standard meaning.
However, there is no value in saying that a fairly widespread usage of the word "subsidy" is incorrect.
It's like trying to pretend that "literally" doesn't figuratively have the same meaning as "figuratively". That boat has sailed and all we can do now is distinguish be between literal and figurative uses of "literally".
There are a range of actions that are appropriate to take when dealing with an unwanted subsidy. If we redefine what 'subsidy' means then the range of appropriate actions changes and we have to talk about what the new actions are. It would be far easier to use the right word, so word and action line up without the need for analogies.
I can't figure out what you are trying to say. Can you give a concrete example? What is the confusion that is added by the term 'implicit subsidies'?
That money they are lobbying for comes from their taxes, especially if it's local roads, so yes, if the trucking company isn't paying for the damage, the taxpayer is indirectly subsidizing them.
If say 10% of a city's budget goes to fixing truck damage to roads without at least commensurate benefit, then it's a more important issue than whether .05% went to cleaning up cotton candy after the town fair.
It also doesn't even have to be about tax expenditure. The healthcare cost of using fossil fuels is realized in private spending on treatment for pollution triggered respiratory illnesses, and the economic cost of climate change effect mitigation, which is already being accounted for in business planning.
I.e. You are demonstrating the exact same error but don't seem to understand what's going on with the use of the word.