So it changes from one region to another. At least in my family we use both "δείπνο" and "βραδινό" for the meal in the evening, but "δείπνο" is usually with other relatives or friends and mostly in formal occasions (like Christmas, New Year's Eve, etc), while "βραδινό" can be even a glass of milk and some cookies in front of the telly. :-)
In the UK the meals are all different depending on where you're from, background, etc. As a kid our 3 were breakfast, dinner and tea. You might have afternoon tea at around 4 pm- cakes, etc. Elevenses at 10 or 11 am, and if you had a late night meal before bedtime it would be supper. The best of course, is the midnight snack, and involves a raid on the fridge and kitchen cupboards...
The sentence "Αυτό είναι το δείπνο" translates to "This is [the] dinner." The translation of "This meal is dinner" would be "Αυτό το γεύμα είναι δείπνο." Note the difference: in both languages, the first sentence contains just one noun (dinner), while the second contains two (meal and dinner).
So, the problem isn't that you didn't understand the sentence properly, but rather that you added in words that weren't in the original. It's analogous to translating "This is a dog" to "This is a cute dog"—while it may be true, it's not really correct to add it. :-)
Here is the Oxford definition of meal: Any of the regular occasions in a day when a reasonably large amount of food is eaten: So, breakfast, lunch, dinner and supper are meals but in Eng. we do not say "this meal is dinner" we say simply "this is dinner" or "this is supper" etc. You can say we had a meal. It is just the nature of the use in English.