Both are correct formats according to different settings
Infopath is the client application rendering dates according to (client/machine/Windows) Regional settings set in Control Panel > Regional and Language Settings > Regional Options tab > Standards and Formulas > Customize * > Date tab in (in Customize Regional Options dialog window) > Short date format* which you can change by pressing Customize... or choosing from listbox options button there. See Fig.1.
Fig 1. Client machine regional settings (used by client Infopath forms)
Sharepoint shows dates according to (Sharepoint server site) Regional settings < Site settings (in Site Administration section) < Site actions, on ribbon, in browser:
- < sharepoint_site >/_layouts/regionalSetng.aspx
which again one can set to whatever settings one would wish. Cf. Fig.2
Fig 2. Sharepoint site regional settings
So, I navigeted in browser to a sharepoint site > Site actions > Site Settings > Regional settings (under Site Administration section) > in Locale listbox changed from
English (United States) to English (United Kingdom) and the format of date shown in browser (used by Sharepoint) became the "correct" one (changed from previous 'MM/DD/YYYY' to 'DD/MM/YYYY', which is what I have set)
It worked for me in Sharepoint online 2010 (Office 365) while the found on internet solutions of setting date format in code did not work
Note that I wrote about machine operating system and sharepoint site's regional settings, not formatting used in Infopath (date picker control) itself
Just to second my guess that the same Infopath form uses different settings.
If it is run in browser, i.e. through/by IPFS Sharepoint Server 2010's Infopath Form Services), then it uses sharepoint settings and when as Filler/client application then local/client machine Windows settings.
For example, in the same form username() in browser shows my sharepoint username (i:firstname.lastname@example.org) and run as Filler/client form the username from client machine Windows (Administrator).