Actually you should say "We have dinner at 7". After "start" you can use both gerund and infinitive, e.g. start doing smth / start to do smth.

They are both fairly horrible, I think. "We have dinner at 8", or just "we eat at 8", might go better. https://books.google.com/ngrams/graph?smoothing=3&corpus=26&year_start=1800&year_end=2019&content=we+eat+dinner+at%2Cwe+have+dinner+at%2Cstart+to+eat+dinner+at%2Cstart+eating+dinner+at&direct_url=t1%3B%2Cwe%20eat%20dinner%20at%3B%2Cc0%3B.t1%3B%2Cwe%20have%20dinner%20at%3B%2Cc0%3B.t1%3B%2Cstart%20eating%20dinner%20at%3B%2Cc0

They mean the same. But the more common way of saying it (US) is: Dinner is at 7. (or Dinner's at 7). If you want to be more formal: We dine at 7.

Start to do focuses more on a single activity, starting point of an activity, denotes more spontaneous actions, unprepared actions, indicating the very beginning of the action rather than its process or continuation. Start to with stative verbs is more preferable: start to feel, start to seem, start to appear, start to sound, etc. Start doing focuses more on continuing activity in the process, denotes prepared actions, not the actual start of the action.

I agree they have the same meaning. It is more common to say sentence # 1