You don't seem to provide any indication of what your 'localtimestamp' is - is that pseudocode? A variable name? A column you haven't shown the definition for?
What data type is 'localtimestamp'? What data does it contain? Pertinent questions as other answers point out, because if it truly is a time stamp then oracle will be converting it to a string for you, before passing that string to to_timestamp() in your final query. Your initial stab at it should just work if the variable is a timestamp, containing a timestamp
Ultimately "date format picture ends" means "you passed me a string looking like '2017-05-17 12:45:59', but claimed it was only 'yyyy-mm-dd'. What was I expected to do with the rest of it?"
Your current final comment on your question "I was hoping to look in the table and see a useful looking time" - that's your query tool's problem. Have a look in the setting of your query tool and change the date format it displays. As has been noted, dates in oracle are stored as a decimal number days since a certain moment in time. If 0 represents 01 Jan 1970, then 1.75 represents 6pm on the 2 Jan 1970. It is up to the end program the user is using, to format the date into something you like.. you cannot "insert a timestamp with a different format" because time stamps don't have a format any more than a number like 1.75 has a format. It is what your query does with it when it gets it out, that gives it the format:
To_char(timestampcol, 'yyyy mm did')
To-char(tomestampcol, 'mon dd yyyy')
These use oracles built in date formatter, that turns that decimal number of the date into a string in the given format; you will see a string.. or you can just write "select * from table" and run it in TOAD and toad will show you the dates according to the format in settings, or you can write a c# program and get a load of date objects out and call my date.ToString("yyyy-MM-dd") on them to format them. The idea I'm trying to get across is that you don't pick the date format on the way in, you pick it on the way out, if you don't like what you're looking at, you have to change it on the way out, not the way in