TITLE: Comparing the existing formulations of universal algebra and their levels of generality
QUESTION [11 upvotes]: I am a newcomer to universal algebra and I just read this (very good, IMO) book on the topic:

Adámek, J., Rosický, J., & Vitale, E. M. (2010). Algebraic theories: a categorical introduction to general algebra (Vol. 184). Cambridge University Press.

The authors most notably treat the case of 1-sorted first-order finitary Lawvere theories (equivalently: finitary monads). This is, as I understand it, the most common level of generality appearing in the literature to deal with (categorical) universal algebra.
Additionally the authors explain the relation between "old-fashioned" universal algebra -- which we'll call equational theories here -- and Lawvere theories, as well as the relation between Lawvere theories and categories of models of Lawvere theories. The quick and informal upshot is that there is a pair of such adjunctions:
old universal algebra, lawvere thies, models
Where $\mathrm{EqTh}$ is the category of equational theories (that is: signatures + equations), $\mathrm{AlgTh}$ is the category of Lawvere theories, $\mathrm{AlgCat}$ is the category of algebraic categories. Algebraic categories are basically defined to be categories of models of Lawvere theories.
The first adjunction connects particular presentations of theories to Lawvere theories. The second adjunction is some kind of Gabriel-Ulmer duality and links theories to their models.
My questions is: what are the existing ways we can generalize this picture? And is there an ultimate framework encompassing all those generalizations?
For instance I would be interested to hear about what can be done to obtain type theories as some kind algebraic theories.
Some examples of generalisations (either on the presentation side or on the Lawvere theory side or on both sides):
higher-order logic, enriched Lawvere theories, sketches, Makkai's first order logic dependent sorts (FOLDS), infinitary theories. I'm sure there are a lot more!
EDIT:
Here is a paper about higher order algebraic theories

REPLY [4 votes]: @focso’s answer and @varkor’s answer give excellent answers covering most aspects of your question.  I’ll just add a little more on extending these frameworks to cover “type theories” — i.e. to include both sort-dependence and variable binding.  This is a quite new and open topic, but a good bit of groundwork has been laid.
The well-established part dates back to the 80’s, starting with Cartmell’s work.  This defines contextual categories, and later various slight variants (categories with attributes (CwA’s), categories with families (CwF’s)…) corresponding to a so-called “generalised algebraic theory” or “dependent algebraic theory”.  This gives a nice treatment of dependent sorts, but not yet of variable binding.  Specific type theories with binders then correspond to extra algebraic structure on contextual categories (or CwA’s, CwF’s, etc).  A good introduction is Martin Hofmann’s chapter Syntax and semantics of dependent types, doi:10.1007/978-1-4471-0963-1_2, un-paywalled pdf here.
For individual type theories, this correspondence works satisfactorily (and has been developed by many authors — I won’t try to give a bibliography here); but until quite recently, it wasn’t generalised to a general notion of “type theory”.  Approaches to that so far include:

Isaev, Algebraic Presentations of Dependent Type Theories, arXiv:1602.08504. This defines type theories directly as certain essentially-algebraic extensions of CwA’s.

Uemura, A General Framework for the Semantics of Type Theory, arXiv:1904.04097.  This in my opinion is the most powerful and best developed approach so far.  It gives a categorical abstract definition of type theories as representable map categories (an imaginative and far-reaching generalisation of Fiore/Awodey’s re-analysis of CwF’s), and a corresponding syntactic treatment (based on the “logical framework” approach).  It’s much more general than either of the other approaches I mention here, and very cleanly set up.

Bauer–Lumsdaine–Haselwarter, A general definition of dependent type theories, arXiv:2009.05539.  A syntactic definition of general type theories, taking variable binding seriously as primitive, and written with a categorical analysis strongly in mind (though not given in the paper). This should correspond to a slight variant of Isaev’s algebraic definition, and to a subclass of Uemura’s.


I very much hope that these and the connections between them will be developed further in the near future, and that this part of your question will have a better answer in five years than it does now!