Check your Merriam-Webster's Dictionary. You'll find:
5 a: a berthing area for small ships or boats, formed by enclosing piers or jetties
b: a dock
or slip for reconditioning submarines that is protected against aerial bombs by a superstructure of thick concrete.
Usage of 5b in US English
dates to at least 1932 (perhaps earlier). Usage of the sense of 5b in UK English
may date even earlier.
Usage of 5a in US English is not well documented by Merriam-Webster, but M-W moved to include it (and M-W has a record
of doing so only because of usage in the US).
The OED ignores the usage of 5a (that's not unusual; OED has a record
of paying little attention to sailing terms).
shows published use of 5a in UK English dating back to 1924 and recent use of 'yacht pens' and the like in UK English since the 1970s.
Merriam-Webster's inclusion of usage 5a suggests that using 'pen' for a 'wet berth' is not unknown in US English. Usage 5b, 'submarine pen', is recorded in both US and UK English.
suggests that 'pen' for a small boat is more common in UK English than US English.
You are free to choose your words! No regulations
Apart from 'berth' you can:
* use a pastoral term (penning your boat in the same way you'd pen sheep or goats; 'pen' can be traced to Old English in 957, but identifying its ancestry beyond that is likely difficult); or
* use a docking
term ('slip' dates from late Middle English of 1467, meaning a stone ramp
built to provide a landing place for passengers and goods; the etymology of 'slip' is clear, with ancestors in proto-Germanic and ultimately proto-Indo-European).