Both sentences are possible, but they suggest somewhat different things.
I finish my jog at 8, before the rain will start.
(Note: I've corrected "my jogging" to "my jog". Alternatively, you could say "finish jogging", without the my, though that doesn't work quite as well IMHO.)
This could work in a science-fiction novel that's narrated in the present tense: science fiction in that it implies exact foreknowledge of the time that the rain will start (perhaps the rain is scheduled by a weather-control system?), present-tense narration as opposed to the more usual past-tense narration ("I finished my jog at 8, before the rain started"). Even in such a context, it would be more usual to write "I finish my jog at 8, before the rain starts" — we don't usually use will in adverbials of time — but I think "will start" is OK to highlight the sci-fi nature of the foreknowledge.
I will finish my jog at 8, before the rain starts.
This still suggests a science-fictional foreknowledge of the rain, but less strongly; I could actually imagine someone saying something like this in a real-life conversation, if they trust the weather report enough. ("Wait, you're going for a jog? Isn't it expected to rain soon?" "Don't worry, I'll be back before the rain starts.")
Edited to add: Incidentally, your question refers to "the First Conditional" and "Conditional 1", but your sentence is not actually using a conditional construction at all. (That said, I don't think there's a true difference between the grammar of first conditionals as in "I'll come back if it rains" and that of adverbials of time as in "I'll come back when it rains" or "I'll come back before it rains." In fact, we can even say things like "I'll come back if and when it rains.")