They're all fine with me as long as game-play isn't fundamentally altered.

As far as being "cash grabs", it's a for-profit industry. Pretty much everything is a cash-grab. All I ask is that value is taken into consideration when it comes to pricing. A straight port of a game from last gen should be cheaper than something that cost money to update.

Well, there seems to be lots of confusion and I'm not sure, too. Maybe because many games are ports with remade elements.

A port is bringing the game (based on the same code) to another platform. The resolution can differ and the textures are identical to the original release or worse (handheld ports for example).

If you use the term "remaster" in analogy to movies and music it should mean that new textures are created from the uncompressed original assets at a higher resolution than the original release. Like a movie gets a new digital transfer of the original film. But the term is also stupid because video games don't even have a mastering in the same way that Blu-rays or CDs have it...

Remake: (all?) assets newly created and not based on the old code.

Super Mario All Stars

And I wouldn't call it a remaster if they had to go in and make entirely new assets. Remasters generally involve bumping the render resolution, and then updating things like lighting, and maybe going with higher resolution textures.  It's hard to use the term "remaster" with 2D games where you're required to visually remake the game from the ground up.

But this whole remake/remaster debate is such a grey area.

As long as they have Achievements, I'm fine with any of them.

Port is just a version from a different console to another and has a different release date? remaster is something like an upgrade port/version to me.
Remake is you change a lot from the game.
I'm okay with remasters as long as it is not a remaster of a game that was released recently or just from the previous gen which has little graphic bump. it is somewhat money-grabbing. But it is just me, and I know some remasters tend to sell well.

Port: Same game, new platform (may contain new content)
Remaster: Same game, presentation and performance enhancements (may contain new content)
Remake: Same story, new mechanics, new assets built from scratch (may contain new content)
Re-imagining/reboot: New game/story, same core ideas/concepts.

Ports and remasters are pretty much the same thing, only different by the time that separates them.
Many multi platform games have differences in assets, resolution, frame rate and effects. We only call them remasters when they come out a lot later than the rest.

It's the same with movies, the first batch on dvd and blu-ray are ports from the movie basically. Second time and onwards it's called a remaster. The process is the same, only technology has improved.

I wonder if Rise of the Tombraider PS4 will be sold as a remaster or a port? It will be a year later, probably with some extras and a few enhancements, If Tombraider definitive was a remaster, I guess Rise of the tombraider will be too?

Remakes: write the code from scratch and create new art assets, new dialogue etc.
I would say different versions of Tetris are remakes. Different code, different assets.
Yet the countless versions of Dragon's lair lean closer to remasters, although I doubt the code has anything in common between Laserdisc and iOS versions.

Tomb raider Anniversary Remake.
Tomb raider 2013 Prequel / Reboot
Tomb raider Guardian of light Spin-off

There was a discussion about this in another thread. Some one wanted to point out that late PC ports almost always release with presentation and visual enhancements. So should they be considered remasters? The concensus at the end was "YES". They do fit the definition because they were made after the original and improvements were made on what was already there. In vast majority of cases, "port" and "remaster" are synonyms. (And lets be honest, if TLOU wasn't a popular game we probably wouldn't be seeing the labeled 'remaster' being discussed so much... Then again, people would still be talking/complaing about 'definitve' editions XD)

Are all ports remasters though? I would argue no. The word "remaster" implies some improvement which isn't always the case. Dark Souls PC is an excellent example. Locked at 720p30, same visual and performance hits that were present on the console version. No improvements were made. Same goes with FFXIII PC, FFIV PC (port of the DS game and even has battles locked at 15fps...) Bioshock PS3 (had some bugs that result in visual downgrades compared to other versions) and some other games I just can't think of at the moment.

This. 

they are all just ports except Resi and MGS examples