New time machine back up doesn't delete files deleted from computer

I am driving myself crazy trying to do a time machine restore on a new hard drive! I am trying to replace my old 500GB HD with a new 250GB SSD. Here's what's happened so far:

-I keep my time machine back ups on an external 2TB Seagate drive, so they don't back up externally unless it's connected, which it hasn't been in a while.

-Haven't done a time machine back up since April, so I connected the 2TB Seagate drive to my 2012 Macbook pro and did a time machine back up by hitting "Back Up Now".

-Installed new SSD drive

-Attempted to restore latest back up; received error that there wasn't enough space. Was only able to restore back up from April, which was annoying because I've done a lot of re-organization and a lot of files were moved and apps not set up, etc.

-Re-installed old 500GB HD, realized I had approx. 350GB used, so I manually backed up photos/etc. to ANOTHER external drive - a 1TB Western Digital - via drag and drop. Then I deleted those files that I backed up off the Macbook and emptied recycle bin. After that, there was only approx. 165GB on the Macbook HD.

-Did another time machine back up. Time machine back up was very quick, and all it said was first "preparing for back up" and then "backing up x out of 859MB". It appears to have only ADDED stuff, but not deleted.

-I stupidly assumed that time machine back ups meant they BACK UP THE SYSTEM AS IT CURRENTLY IS, so I re-installed the new SSD again

-Still getting error that there is not enough space on the SSD to restore latest time machine back up.

I have read other threads similar to this, and most of the fixes involved issues where they manually excluded files from the back ups. I did not. I deleted these files so that the time machine back up would only be of the 165GB of data that is currently on the HD. Time machine did not appear to create a FULL NEW back up; instead it just does some weird thing where it "updates" files on the back up and labels it with a new time. This is my assumption based on the things I've seen and how quickly it does a "back up", and how it states "backing up x out of xMB" - it should be removing files, but it is not. I do not know how to check the size of the TM back ups.

So basically there were large time machine back ups from when I had about 350GB used on my Macbook, and it appears each new back up is only being updated, but no files are removed from the back up even though they were removed from the old hard drive. How do I get time machine to do what it's supposed to do and do a full, new, smaller back up of 165GB which is the amount of data I currently have on my old HD, so that I can transfer that 165GB of data and all my current apps/settings onto my new SSD?