Which dialogue format do you prefer?

I prefer Format 2. I think it's easier to read. It also takes up less space. 

I prefer format 2. It suits the style of a story and connects the words spoken to the speaker a bit more. 

Actually, what you have formatted, either style, is more for a movie script.  Try using John said, "I laughed so hard that I cried!"

New line before the other person replies, "Too funny!" Alisa said.

That's what I was doing at first. I decided to change to the script format because I find for me when I read back on the dialogue it's easier to read the dialogue that way as if it's more like up front and direct kinda like reading a chat on an MMORPG. Also, what I'm doing with the changes to my story is I'm telling it as a blog kinda like a reality show which is what I intended to do from the beginning. I found using the story format dialogue makes it like a written story rather than telling the story as a blog and reality show. In other words, I just found using the story dialogue format to be less real and direct than what I am trying to go for if that makes sense. I guess it's more correct to say that this not a written story like a novel, but a blog or diary telling the events that happen as each day progresses as well as sharing some of the conversations that took place.

Say for example this is just a made up description of the blog:

The Adventures of John and Alisa
This is a diary about the adventures that me and my girlfriend Alisa share. Some of the things you'll find here are the details of our adventures and some of the conversations that we have.

Example:

Below is another example of John telling an event that happened, and then John sharing a conversation that he had with Alisa:

Today we adventured through the darkest of the forest. We encountered a few werewolves, but we were able to defeat them without much trouble.

John: Those werewolves were ferocious, but they were no match for us.

Alisa:  *Tickles John*

John: Hey! Stop that! *Laughs*

Alisa: *Chuckles* Yeah, no werewolf is a match for us!

John: Well, let's continue on. I'm sure that fortress must be around here somewhere.

The problem I'm facing though is I'm struggling to decide if I should use format 1 or format 2 for this type of dialogue. I find there are both benefits and drawbacks to both formats. I just wanted to see what others preferred. I think I'm leaning on going with format 2 though.

My vote is still for Format 2, but if you add actions (ex. *chuckles*) to the dialog it may get a bit confusing/distracting because of inconsistent verb tenses. It's hard to tell if the conversations are taking place in the present or the past. They use present tense (ex. *tickles*, *laughs* ...), but all the text that introduces the conversations is written in the past tense ("We encountered a few ...").

Format 2

I can see what you mean. What about if it is looked at in this perspective. Say in the intro paragraph where it says what they did, but then the conversation switches over to a present tense as if they were talking and it's being captured.

Example:

Today we adventured through the darkest of the forest. We encountered a few werewolves, but we were able to defeat them without much trouble.

So, John explained what happened today, but then the the conversation goes to present tense as if it's happening right now and he's recording the conversation as he goes.

John: Those werewolves were ferocious, but they were no match for us.

Alisa:  *Tickles John*

John: Hey! Stop that! *Laughs*

Alisa: *Chuckles* Yeah, no werewolf is a match for us!

John: Well, let's continue on. I'm sure that fortress must be around here somewhere.

So the conversation is almost like a transition into the next blog post. I don't know lol! These are just examples. In my blog, I'm trying to capture it in a way where it feels real and live like it's happening as it goes, but at the same time using an explanation paragraph of what happened so far in that post etc.

If you want it to feel live and as if the reader is in the action, what about just using present tense throughout? I think either way will be okay, but present tense has a tendency to make things more exciting. (ex. Today we adventure through the darkest of the forest. We encounter a few werewolves ...)

Yes, I think the disclaimer will help a great deal. Either format will do, putting it that way. I think I prefer the second one, over the first example, though.