While the period of time before the fusion of law and equity in 1873 is quite long, and there were many changes to the law during it, I think it's reasonable to look at the earlier part of the nineteenth century as the heyday of equity practice. That was a period when it was routine, reasonably prestigious, and formalized - much earlier, it is harder to identify a systematic approach to rights of audience at all.
I picked out A Popular and Practical Introduction to Law Studies by Samuel Warren (London: Maxwell & Son, 1835) to see what advice he had to give to people entering the profession. This book accords with my general sense of the shape of the legal system at this time, and I have no reason to think that it is wrong or unrepresentative. Warren has many opinions about the various career paths available, and what training is necessary or helpful.
At the time, the "central" courts included several courts of common law in Westminster, plus the Court of Chancery (an equity court), the Court of Exchequer (which used both common law and equity), the Court of Bankruptcy (a specialist one, but mostly resembling equity practice) and others. The shape of the legal profession formally included -
- "attorneys" who acted in common-law courts, but didn't argue orally
- "solicitors" who did the same in equity
- "proctors" who did the same in ecclesiastical courts
- "barristers" who could argue orally in all secular courts except the Common Pleas, including the Court of Chancery
- "serjeants-at-law" who were a special sort of senior barrister, in the Common Pleas but also other courts
- "advocates" who were barristers in the ecclesiastical courts
The sub-question about law firms does not match how barristers organized themselves, since they were not employed in firms. They would have been in chambers with colleagues, and part of one of the Inns of Court. Litigants would not engage them directly, and solicitors would know who was who, and who was good at what.
People appearing in equity courts were "barristers", but with a professional speciality
In order to appear in the Court of Chancery, someone would have to have permission from the court. That is distinct from whether they had the same permission for any other court, such as the King's Bench. Any such person would have to be a barrister to begin with, but there was this extra procedural step in order to gain rights of audience in the specific court. This was simplified in the 1840s, so that there was a common right to appear before any court of equity, and a separately obtainable right to appear before any court of law. These were naturally merged into a single right once the systems were combined. In the earlier period, the credentials are the same (the difficult part is being called to the bar at all) but not everybody bothered to obtain rights in every court. From Warren, in his time,
According to the Law List, nearly five hundred of the two thousand five hundred members of the Bar, are Equity Counsel: but it is believed that not more than two-thirds of the number are actual practitioners. It may be added, that the leading Common Lawyers appear much more frequently in the Equity Courts, than the members of the latter present themselves in the Common Law Courts.
These barristers might be called "equity barristers" or "equity counsel" but that is not a distinct title, just a way of indicating their area of work and expertise. In the same way, we might talk about "criminal barristers" but that just means a barrister who specializes in criminal work, not a separate class. The term "equity draftsman" is very common, because writing pleas was a difficult and critical part of practice; this function was almost always done by a barrister.
The barrister/serjeant distinction was important at times, because there was an order of precedence at Chancery hearings. Everybody who wanted to make a motion in any active case would queue up in order, on a sitting day, and wait their turn. The Solicitor-General had the first turn; serjeants preceded barristers; and King's Serjeants or King's Counsel preceded others. But "barrister" is the generic term, or something like "equity barrister" to stress the specialism.
Some remaining points
On the other side of the profession, Warren says that attorneys and solicitors were mostly in fact the same individuals, who would be duly qualified to act in any of the courts. But a large practice would include partners and juniors who specialized in different areas, so some people would spend all of their time on Chancery work. Outside of London, this was less common. Equity cases on circuit were rare, and so there were few specialists.
In terms of training, all of these people would have been exposed to learning about law and equity at some level. But formal legal training was very theoretical. Actually drafting pleas and presenting them is an entirely different set of skills, learned "on the job". Because Chancery practice was procedurally different from that of other courts, there was a wide gap between being technically permitted to appear in court, and being actually good at it in career terms. Warren's advice to students is that they try to gain exposure to the various branches of practice in order to find out what is the best fit. So while people enter with the same sorts of formal qualifications, and could in principle act as counsel in many settings, they did differentiate themselves - as they continue to do today.