The second form you posted comes from the RKATS article
Expenses and Reimbursements. In that training video, it says (at about the 4:50 mark): "Each ward should create a form to gather basic information for each expense. The format of the form is not as important as the information it requests." In every ward or stake where I have served, the wards have done exactly that -- created their own form. Some were very detailed, and some were simple, but all had fields for the same required information. The image from that training article is simply an example, so if your ward has a form that it is using, there is no reason the example would replace your form.
The first form you said you "just received from your area". It seems like a reasonable form, but I can't answer your question about whether it would replace your ward's current expense authorization form. That would depend on what communication you received from your area leaders. Were they supplying it as an example, or as an optional form that you might use, or are they telling all units in the area to begin using it? If the answer is not clear, you might want to ask your stake or district clerk, who should know those answers or can communicate with the area leaders to find out for sure.
Questions that can benefit the larger community should be asked in a public forum, not a private message.
