First of all you need to realize that there isn't much "objective" to work with...
Νοτ when it comes to Mage's classification of magick as coincidental or vulgar, at least. In almost all editions of the game (M20 included) there are many arguments for either side in almost every Effect. Sometimes there are explicit declarations (Prime-charge energy rays) or the situation is stark clear on vulgarity (throwing a fireball from your hands) but even then there are mitigating circumstances (Reality zone, player argues about the gas explosions earlier in the evening making the fireball coincidental since you're near a pipe yada yada...).
Throughout the history of the game this has been embraced, to an extent, by the community as a noteworthy aspect of the game's tone ("Reality is subjective") and there have been few attempts to put a finer lines on coincidental/vulgar, leaving the matter to the ST's hands, for the most part.
Hence it is a rather common saying among my groups that "A Mage character's power is very dependant on the ST's style and limitations". The exact same character sheet might be either a threatening reality bended with few limits that can wipe other supernaturals of equal experience, or it can be next-to-useless when it comes to magick because almost everything requires higher spheres and/or incurs Paradox when it has any actual impact under your ST.
As a result I don't think there can be a an objective definition.
How would you objectively define just how "loosely" you adhere to HOP or HAP?
Simply because the game is not designed for that. So...
What's the next best thing you can do to have a formal/objective approach to an effect's vulgarity?
Essentially you need to decide on your angle (which you have, something between the HOP and HAP paradigms) and try to workaround the possible ambiguity issues:
- Declare your choice of hybrid HOP/HAP to your players to give an initial foundation on what constitutes a vulgar or coincidental Effect.
- Decide on theme/setting. Do you want a game where conjuring fire in your hands or voices of spirits can be treated as ordinary? "Yeah, that's cool. Where did you hide the speakers though?" Or something that's close to our world's solid reality? watches a guy open his bag and pull out the exact weapon they need to slay the supernatural creature, for the third time in the night. Runs away with widened eyes
- Examples. A very important and direct solution is to write down and share multiple examples with the group on what would be coincidental and what vulgar. Furthermore, you could add parameters to each one. Things like reality zone, ways the Mage describes the situation or other circumstances that could change the example's vulgarity.
- Define the extent to which a player can affect the narration for his spells in the context of a scene and if mundane rolls can be used for that. Saying that the audience is amazed by your performance or whether they can claim the stage even has such mechanisms around so that the witnesses don't consider the flames coming from the stage odd since you're obviously some A-Class rock star could make a significant difference on the vulgarity from the ST's perspective.
- Finally, bring the players and Storyteller on the same page. Make sure they all understand (in most of the cases) the way she treats their character's abilities. 1) What the "default" vulgarity status for various Effects is 2) Which aspects of the story's scenes are up-to-debate with reality's rules and can be influenced by the players as part of their casting (is summoning a cab to get somewhere fast just a coincidental Correspondence effect or a vulgar one that requires Mind+Entropy as well?) 3) What factors can, potentially, change the vulgarity of a spell, based on how your game treats reality.
And of course anything extra that your group deems worthy...
It should be noted that Mage is a game where the players of many groups, usually, need to utilize their out-of-game reasoning and innovation capabilities to come up with a way to deal with the castings in an almost problem-solving manner. If your group dislikes this approach it would be perfectly reasonable to follow the Golden Rule and convert those player attempts for various coincidental goals into game rolls on behalf of their characters abilities.