You are looking to trademark the name. You will need to register in each country, individually(maybe all in 1 in EU), where you want the mark. Unless you have an extra $100k to spend on lawyers, i would suggest just trademarking in your home country and major market.
I’ve had a lawyer tell me in the case of my lottery brand, i don’t need to file for a trademark. I created the site and have operated with the name since 1998 and there is plenty of evidence. Based on that, how could someone demonstrate that they were using the name first.
Finally if you’re going to try to trademark “BetOnlineNow” for example, could you defend it in court, if say betting online is illegal?