- Did lawyers have a broader sense in Shakespeare's time?* Sort of.
Emperor Claudius of Rome legalized the profession of law, allowing men to charge fees for legal advice. Prior to that, they could only serve in a lesser role known as advocate in regards to Roman courts allowing a "friend" to testify on the accused's behalf.
This separated the practice of law into two main groups, clerks (tabelliones), who were essentially notaries in the modern sense, and advocates, taking on more of the professional sense of today.
In general, clerks were barely literate, but could still draw up contracts in the realm of civil law. Advocates were fully literate and often enrolled in the furthermemt of Canon law. By the sixth century, advocates had to be approved by a court and joines up in the local bar in order to practice law. These were essentially lawyers in the modern sense.
The legal profession disintegrated during the Dark Ages and reappeared in the twelfth century primarily in ecclesiastical issues. A course of study was established of about four years.
Henry VI (reigned 1422-1471) was an interesting case. He initiated two universities, King's College (Cambridge) and All Soul's College (Oxford).
However, during the late years of Henry VI, King Ferdinand of Spain made the practice of criminal law illegal, possibly as a precursor of expelling Jews and Muslims from Spain and as rhetoric for propaganda in regards to heresy charges and executions supported by the Bull of the Spanish Inquisition (1478 by Pope Sixtus IV).
The ancient roles of clerks and advocates were still separated based on education and acceptance into the local bar. I don't think we know enough about the culture in England at the time among playgoers to know if there was a distinction made by the English between the two. They both charged money. They both practiced law. Lawyers that were in the bar charged a lot more. I would liken it to the difference between MDs and DOs (in America) - nobody outside the medical profession knows the difference.
I'm hardly a Shakespearean scholar, but it seems he liked multiple meanings in his prose. It's possible that 1. The lay population did not know the distinction between clerks and lawyers, clumping them all together; 2. Shakespeare, among others, was aware of the propaganda in Spain denouncing lawyers of all types (a profession with a high percentage of Jews and Muslims) in Spain; 3. complicated jest designed to make the audience think; or 4. a backlash against the wealthy and educated following the creation of new universities in England by Henry VI.
Note: There are a ton of references here, mostly re-referenced from The Later Roman Empire by A. H. M Jones. I've spent enough time on this for one day, so you may downvote me if you think I'm completely talking out of my arse.