The "rule of thumb" is if the person spoke Finnish, you write the name in Finnish (Matti Juhonpoika) and if the spoke Swedish, you write it in Swedish (Matts Johansson). If your family was from Maxmo (Maksamaa in Finnish), odds are that they were Swedish speakers. Please note that Lisa and Lovisa are -dotter, not -son (Andersdotter, Eriksdotter). Patronymics were used, but mainly in Western Finland. Last names didn't become compulsory until the 1920's, so those are usually found in Eastern Finland. There are exceptions, of course. Some people like to use the name of the birthplace as a last name, some use the name of the farm/croft they settled on. Some enter the patronymic in the surname field. I don't think there's a "right" way to do it. Just settle on one way and explain your choice. :)
The national archives Digital Archive has (most) Maxmo records digitized and it's free to use. Here's the family when Lovisa was born:
http://digi.narc.fi/digi/view.ka?kuid=5412606Erik's parents:
http://digi.narc.fi/digi/view.ka?kuid=5405100Lisa's parents:
http://digi.narc.fi/digi/view.ka?kuid=5405234The Digital Archive can be accessed here:
http://digi.narc.fi/digi/?lang=en_USBut it's very cumbersome to use and most people prefer this resource to navigate the site:
http://en.digihakemisto.appspot.com/Finland's Family History Association also has free (and members only) records online, but they don't have any Maxmo records atm.
http://www.digiarkisto.org/sshy/index_eng.htmHTH