What makes a consulting firm a "true strategy firm" and a boutique an "elite" one?

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These are both good questions, and unfortunately the answers to them are unclear. Consulting isn't as easy to organize and split up compared to banking. Hopefully my thoughts below help. FYI this answer applies to management consulting, specialized types of consulting like restructuring or economic consulting will have different players (and aren't worth mentioning since strategy consulting usually refers to types of projects under the industry of management consulting)

"Strategy" is a pretty nebulous term in consulting, but broadly you can think about strategy questions as higher level questions about what a company should be doing, vs more tactical questions about how they get there. There are lots of gray areas and ways that you can argue a certain type of project is or is not strategy and there are lots of them, so I won't try to be exhaustive, but types of work often include things like which markets should we invest in / pursue, which product types should we invest in or divest from, private equity due diligences, and pricing work / revenue growth management

Regarding which firms do strategy work, ultimately there aren't that many. Your best bets are MBB (McKinsey, Bain, BCG) and the T2 ('tier 2') firms (EY-Parthenon, Kearney, LEK, Oliver Wyman, Strategy&). In the Big 4, Deloitte does a decent amount of strategy work but has declined significantly over the last 10 or so years, EY doesn't do much outside EY-Parthenon, KPMG has a small but growing strategy practice, and PwC like EY does most of their strategy work through their Strategy& brand

Now is when it gets confusing. Outside of MBB/T2/B4 there aren't very many clear labels or groupings, and people will often throw the term "boutique" on everything. There are larger firms that are known for doing solid strategy work in these groups such as Simon Kucher (pricing work), ZS (pharma and life sciences), and a number of other healthcare players. Then you've got much smaller firms that the label of boutique is more appropriate for, such as Keystone Strategy. Unfortunately answers about this longtail of boutiques are hard to come by, and I don't think I know enough to give a good who's who

Lots of companies will throw the word strategy in as you mentioned. It can often be outright misleading, sometimes it can be based on a reasonable stretch of the word strategy, or other times it's true but the mix of strategy vs non-strategy is not as high as you're led to believe (I'd put Accenture Strategy here based on what my colleagues who used to work there tell me)

I'd put the WITCH companies (Wipro, Infosys, TCS, Cognizant, HCL) in the group that in reality does very little if any strategy consulting

Summary so far, your mix of strategy work is best at MBB, then T2s, then you can get some in more specialized areas at certain boutiques but wouldn't count on it as either (a) scope is limited or (b) strategy mix is low. This industry is very reputation based, and in general when companies need guidance on their most important directional questions the list of firms they work with is pretty top heavy

Now onto your questions -

1. Who should you work for: MBB > T2 > others, and you can debate the order for the others. In my opinion, how much strategy work they do is only one factor, other factors are comp and exit opportunities (i.e. brand strength). Two implications are (a) working at a boutique that does strategy work won't be as helpful for exit ops as working at e.g. Strategy& since it doesn't have as established and recognized a brand, and (b) you're probably better off at a big 4 than doing "strategy" work at a place like Infosys since Infosys' reputation (and by extension exits) are within a very different focus

2. What is an elite boutique: that term is not used at all in consulting, and boutique itself is a pretty sloppy catchall for everything that's not MBB/T2/B4 (can a company with 10,000 employees be a "boutique"?). This stems from the fact that outside of the big 4, consulting firms don't really group themselves like some banks or PE firms do (I'd argue that this is because the types of services offered under the umbrella of 'consulting' are usually much more varied between firms than those industries)

Edit: One other difference to add for comparison vs banking. In IB, as I understand it elite boutiques can sometimes be just as or even more desirable to work for vs bulge brackets, but that is not the case in consulting. I cannot think of any not-deeply-individual instances where I would recommend an entry level hire go to a non-MBB firm if they had the competing MBB offer