There is no definite answer as far as I can see. Most of the namings are from different games or communities. I think you can use the following information as a guideline:
Targets damage towards enemy carries primarily and uses their skills to get to that carry and tie them up.
Does some damage, but is primarily there to tie up anyone who is trying to mess with their team's carries. They are primarily used as an additional sink and to pick up enemy bruisers who make it past the tank.
Pretty much an in between of both bruiser and off-tank.Can take a hit and not have to burn flash to live. They can put out respectable damage as well but not enough to really be a threat to most tanks.
A lot of characters can fit into any of these roles depending on how they are itemized.
Jarvan with Trinity Force is more bruiser. Jarvan with Frozen Mallet and Randuin's is more off-tank.
An other explanation I found on Reddit:
A bruiser is built to be in the enemy's face AND deal consistent
threatening damage. To balance this, they typically have little CC,
but often have damage soak abilities like shields or heals. Good
examples for me include Udyr, Nocturne, Irelia, Renekton, Vi, Lee Sin.
A true tank will have a kit not built for damage, but exceptionally
good in survivability and often built to scale off of defensive stats
like health or armor. This allows them to build fully defensively and
be a mild damage threat, but the primary reason they can't be ignored
by the enemy is their CC. Malphite, Sejuani and Shen are a few good
examples for me.
Offtanks are basically the in-betweens of these. They'll have some of
the facets of each side. Often they can do okay on fewer items thanks
to a big tank ability plus natural damage or threat level in their
kit. Leona, Alistar, Nasus, and Jarvan are some of the commonly played
champs I'd put in this category.