Big money can aim for piles. For instance, a big money/black market/fairgrounds deck may aim for piling out on fairgrounds, havens and schemes.
I think a better strategy is to define some qualities that all decks have and then to look at where various types of decks fall.
For example every type of deck in dominion has a way to get VP - either by dint of having chip gain or by purchasing VP cards with either treasure or through the play of action cards (e.g. KC/Mandarin gives a province, so too does KC/bridge/feast, witch gives you effectively +1 VP when you play it). I like to call this payload. Every deck has some type of payload, big money strategies tend to use more treasure for their payload; conversely engines tend to use more actions and chip decks revolve almost entirely around the actions. One way to measure this is to look at a typically hand and say what percent of VP acquisition came from actions and what came from treasures (e.g. two festivals and two silvers is .5; one festival and two golds is a .75). Chip decks tend to be very near 1, engines can approach 1, and money decks tend to approach zero (BM/smithy is 0).
Another shot is deck size. Bloat decks, like gardens or p. stones, are way off to one side wanting absolutely huge decks. Big money decks fall into the middle of "little" trashing and high card count. Engines fall further in size, though some examples (like HP, minion, etc.) can be a bit on the large size. Golden decks (e.g. bishop, KC/KC/monument/monument/trasher, e.g KC/KC/goons/goons/watchtower/watchtower/trasher) are, obviously down at the smallest end of size.
Another useful metric I find is action density (per hand). This just measures how many actions you play in a typical hand. For straight BM, it is 0. For most BM, it is <1. For golden decks it is = 1. For engines it is always > 1.
Lastly, I like the metric of deck progression. 1 denotes playing all the key cards in your deck. 0 denotes moving forward just 5 cards (due to attacks deck progression can actually be negative). Again engines tend to have high deck progression, golden decks tend to be near 1, and BM tends to be near 0.
With these metrics we can talk about things a bit more intelligently. Something that has medium deck size, low deck progression, low action density, and low action payload (or high treasure payload) - most likely it will play like some version of big money. You can call it a "rush", but the hand to hand dynamics are going to be pretty much play a card, buy a card, and progress 5 cards forward.
Hybrid cases would be those instances where things are not quite so simple. For instance a counting house, inn, pawn, warehouse setup would have a low action payout, but have a high action density. Its deck size may well approach bloat, but it plays like a (highly unreliable) engine and has completely different play/buy rules.
Also, something like KC/scheme/B-crat uses lots of money, has a bloated deck, and makes low deck progression ... but it does play a lot actions each turn and plays weirdly. In fact, I think there is room for a special class of decks that I call top-deck control decks. These tend to be larger than most canonical engines, play way more actions than most money decks, and have highly variable payout mechanisms - the big thing is that they tend to setup a dynamic where you can always play a strong combo which may or may not make it through the deck. For instance, watchtower/border village/council room will tend to leave you a strong combo on deck top and with a high enough treasure density you can snag both a province and more engine components.
Likewise, I could see a combo-deck as a separate class. KC/Bridge isn't an engine - you aren't making any special deck progression nor is your deck smaller. However, you are playing more actions and your payout is extremely tilted toward actions.
I think a far better definition process than saying "what is an engine?" is to first delineate some metrics and then say "engines tend to fall around here" so we can not just "is this an engine?" but "how canonical of an engine is it". The important question about classifications is that they inform how you interact with objects of a classified set. Basing definitions off metrics allows for this to be self-evident.