This question is hypothetical.
If I am an independent contractor doing work for a company in my home country while located abroad (working via internet can be very normal without appearing at an office).
In such case, will I need to apply for a tourist visa or for a working visa? Does the answer depend on the visiting country or my home country? Are there any types of legal issues I might need to worry about?
It does in any case depend on the country where you would be when performing the work as every country is basically free to decide who can enter, what type of visas it offers/requires from foreign citizens and what they can do while in the country. Even when the rules are somewhat similar, they are defined independently by each country's laws and not by some overarching international norm.
Similarly, whether your “home country” matters depends on the destination country's laws. Some might provide different rules for citizens of some countries but that's still up to them (and to international treaties like those that created the European Union). Your country of citizenship certainly cannot grant you the right to go or work anywhere else in the world on its own accord. What it can do is free you from the obligation to pay taxes if neither your client/employer nor yourself are in the country (or not, in the case of the US) but that does not mean you have the right to work anywhere else or wouldn't owe taxes in some other country (taxes would probably be the next major legal issue to consider after work permission and visa regulations).
That said, even if the details differ, you can find many commonalities. One of them is that there is often a kind of implicit hierarchy between more restrictive but easier to get business/tourism visas on the one hand and work visas on the other hand (with some exceptions like Saudi Arabia, where tourist visas aren't available at all). So working is typically forbidden without specific authorization/visa/permit. If the law does not explicitly make a distinction, there is no reason to expect that working remotely makes any difference and you do need to have the right to work to do it.
Thus, the notion that “visa regulations haven't caught up” appears to be mostly wishful thinking as local laws apply fully by default, wherever you are, even if they are restrictive or inconvenient. Of course, if you are just using your laptop, don't stay very long and your clients and bank account are all out of the country, you can often get away with a lot but that does not necessarily say much about the law itself. Anecdotes along the lines of “I did it and nothing happened” do not really show otherwise (after all, people steal stuff everyday and often get away with it too but that's still forbidden in most places).
People regularly talk about some country (e.g. Georgia) allowing foreigners to work on a tourist visa (or some variant of this claim) but as far as I can tell it's mostly a case of lax enforcement and I am yet to see solid evidence of anything like that, beside perhaps Svalbard. Elsewhere, people who tell you that it's OK are just working illegally and getting away with it, nothing more.
Also, stating the obvious but “work” isn't “tourism”. If something is called a “tourist visa”, it's probably for tourism. If, on the other hand, a cheap short-term visa lets you work without limitation, it's not a “tourist” visa and it makes more sense to call it something else. But in most places, there is no such thing as a regular visitor visa allowing paid work.
