What are some electronic payment systems available for use by foreigners in India?

Warning: outdated answer

This answer was written during the 2016 Indian banknote demonetisation crisis, and many parts of the answer was written with that in mind. Moreover, there is extremely rapid technological and economic advancement in India, so everything else in this answer is also likely to be less and less accurate as time passes. I no longer live in India, and I will not update this answer as things change.

widely accepted for daily purchases

This doesn't exist in India. There is lots of talk of starting to use systems so that people can give and recieve payments with their smartphones, but nothing has become widespread yet.

In larger cities and towns, some of the larger shops and businesses accept payment cards, but the great majority of businesses do not. This becomes even more true in smaller towns and rural areas, in which cash is typically the only way to pay.

One of the reasons for this is that Indian banks impose fees on card payments, and the business has to pay them. Of those that do accept cards, some don't accept foreign cards.

I use PayTM myself. I think you do indeed need an Indian number to use it. And you need an Indian payment card or internet bank to load money into it. You might be able to load money into it at an office by paying cash, but that would defeat the purpose (cash being hard to get.)

Moreover, PayTM is not very widespread as a payment method in shops. It's slightly more common in big cities, but not really something you can depend on.

My advice: Bring a foreign payment card (VISA or MasterCard/Maestro) with low fees on it and use it when you can. And get as much cash as you can somehow. You simply can not travel in India without cash. You can also bring some US dollars with you, to be used when paying large amounts. It might not be that hard to convince people to accept them for payments, given the current situation here. You might not get change on them, though. But they are great for emergencies.

Yes, getting cash is tricky these days. But ATMs are starting to open now. You may have to stand in line for a while when you find an ATM that is open, but it's worth it to get the cash. A friend of mine with a German debit card stood in line the other day. There was a limit of 2000 per person, but she actually managed to do 8 withrawals and thus get 16.000. Appearantly the system didn't stop her from making multiple withrawals since she had a foreign card. For more advice on getting cash, see this related answer.

In India, if you have a valid and accepted ID proof (your Passport), you can get a working SIM within a few hours especially Airtel. Airtel is one of the leading operators in India so connectivity with that would not be an issue. There are other options as well that are mentioned here: What is a good Indian SIM for south to west coast (Kochi to Goa)?

Once you get a working mobile number, you can create a Paytm account for that number within minutes. Payments using Paytm are accepted almost everywhere in cities and towns. If you go hiking or to a remote village, then it's quite possible they won't be accepting Paytm as well. Also, payments using Paytm require a running data connection, which may not work in remote areas as well.

Loading your payTM account with money can be done electronically using credit cards. However, Foreign credit cards are not supported as per the PPI (Prepaid Payment Instruments) guidelines issued by the Reserve Bank of India

For Taxis, you can use payTM for Uber. Given the limitation of putting money in payTM account, I suggest you use your existing Uber account. If you already have an Uber account linked with your credit card (doesn't matter which country you are living in) then you can use the same Uber account in India as well. The payments for that will be deducted from your currently linked card in INR and foreign exchange charges will be applied.

Talking about Ola cabs, it supports Ola money, the last time when I checked(in February) it did not accept international credit cards.