County pros had winter jobs even in the nineties. Andy Caddick used to paint the fences at Somerset if he didn't go on an England tour. Did this mean he was not a professional cricketer? Of course not. There are hundreds of salaried cricketers in the English system and only about thirty play cricket all year round.
You get paid an agreed contracted salary to do something? You are a professional. Miner, plumber, nurse, cricketer.
The issue is this. Was Andy Caddick doing his cricket drills and training during the offseason? How long were these offseasons? Early day pros simply went off to other jobs or games like football to earn their living. Most of them. And they played little or no serious cricket for months at a time, and this was repeated year after year. They may be pros on paper, but how do they compare with modern pros?
Take India for example and the barriers you need to cross to get into test team India. (The numbers are just for illustration, may not be correct)
1. School/local club team level, some 100,000 strong pool
2. District level, some 25,000 strong pool
3. Inter-district level, some 5,000 strong pool
4. Inter-divisional level, some 1000 strong
5. Zonal, some 200 strong
6. National team, some 30 strong
If you make it to 4,5,6 (maybe 3 also) you can be called pro in theory, who will get a meaningful salary, match fees etc. But if u need to remain a pro you need to train several hours a day all year to stay competitive (even during offseasons), even if you have a job outside cricket. Else you may go down the scale and disappear from the pro circuit in no time. Hanging onto 5,6 categories these days requires elite level training which cannot be sustained if you have other job obligations and are a part time cricketer.
Was it like this in 1920? I doubt it. The talent pools were much smaller back then.