The answer to your question is no. "Packing our bags, we left for home" has a completely different meaning from (2). The phrase "packing our bags" is progressive. That means the action is happening now. In "Packing our bags, we left for home," it means that you are leaving for home and packing your bags /at the same time/. In other words, you do the first thing while also doing the second thing. For example, = "(while) Eating the granola bar, Sara jogged through the park" (Sara ate and jogged at the same time) = "(while) Walking to the bus stop, Jaime played Tetris on his phone" (Jaime walked and played Tetris at the same time) The construction "having [verb]ed" means you finished [verb]ing, and then you did another thing. For example, = "Having brushed my teeth, I went to bed." (First, I brushed my teeth; after I finished, I went to bed. = "Having buried the body of his last remaining relative, Kelly packed everything he owned into a suitcase and left to travel the world." (First, Kelly buried his family member, then he traveled the world.) I hope this helps!

"Packing our bags, we left for home" may have a slightly different meaning to the other two sentences, meaning: "While we were packing our bags, we left for home." This would be a strange, yet not impossible, situation as usually you pack before you leave.