1000 ELO. Am I now an intermediate?

For 99 percent of the people in this world, your statement is completely wrong.

I think there are actually three grading scales. Non-player, amateur and professional.

To me, it must be taken into consideration that most players in an internet site are equivalent to casual players from the time when there was only OTB, and those weren't rated, or couldn't even be considered "proper" chess players in certain way (I've met many players who didn't even know the rules). But everyone gets a rating in an internet site, and this distorts the evaluations discussed in the op somehow.

Gone are the snobby days of otb chess.  It is no longer a gentlema's game.  Now the rabble play. 

The old titles are meaningless.

There is simply a continuum of people, within a certain range you have an actual competition.  Taking all the people who know the rules, that is a lot of people.  That is a lot of really bad players.

It seldom was a gentleman's game.

I think we can all agree the internet has changed the game and FIDE or any other chess organizational rating no longer applies, at least outside their organization.

Forget about FIDE, titles and ratings. A player who displays the skill that takes getting 1000 points here is a beginner.

Been playing rapid 10 mins for a while and was stuck at 700 basically the entire time. Started doing more puzzles and reviewing my games more and have now shot up to 1000 elo. 

Can I now put "Intermediate Chess Player" in my instagram bio?

I would say yes 

You are crazy.  A beginner is at 400 or less.

We need a new title, then, between Beginner and Intermediate.

How about Pooch? or Patzer? Duffer? Munchkin? Thud-and-Blunder-Boy?

Fish? I like blunder-boy even though it kind of hurts.

You can simply be bad and not a beginner.  Some people can't see more than 9 squares, others just move too fast or hope the other person plays poorly and imagines it it a good move.  Neither of those two people are not necessarily beginners, just are bad.

Most people are beginners their whole lives. It is a consequence of playing a game without ever learning anything about it. It’s true of Monopoly, too. Most people are lousy at the game.

Beginner implies just beginning, not being bad.

To say beginner just means you are terrible requires a name change.  It needs to be changed from beginner to terrible.

What word other than terrible can you use for someone who never gains proficiency at something they’ve done a long time?

Define proficiency in chess.

idk

Proficiency means just getting by.

That’s a good topic that deserves some thought. 

I would try to list some basic skills that constitute proficiency, such as knowing well some elementary checkmate patterns, some basic opening principles, having the ability to spot and employ common tactical ideas, …

I don’t have a clear definition ready to hand, but will give it some thought.

My Scholastic Awards have a list that might come close. It was created to lift youth chess players to a level that was competitive in a scholastic chess context. 

Scholastic Chess Awards
Pawn: the Pawn Award recognizes that the recipient knows how to play chess (and should be able to teach another).
1. Sets up board (light on right) and pieces correctly.
2. Demonstrates basic movement of each piece.
3. Demonstrate and explain castling.
4. Demonstrate en passant.
5. Demonstrate ability to recognize checkmate (complete “Pawn Award: checkmate in one” worksheet).

Knight: the Knight Award recognizes that the recipient has learned certain fundamental endgame and checkmate skills.
1. Previously earned Pawn, or achieve a NWSRS rating over 500.
2. Demonstrate understanding of checkmate of lone king with heavy pieces:
* queen and rook,
* queen and king, and
* rook and king (each from two random positions selected by the coach).
3. Demonstrate understanding of “fox in the chicken coop” pawn promotion technique.
4. Complete “Knight Award: checkmates and tactics” worksheet.
5. Demonstrate ability to read chess notation.

Bishop: the Bishop Award recognizes that the recipient has developed skill in coordinating the chess pieces, including honing his or her checkmate skills.
1. Previously earned Knight.
2. Force checkmate of lone king with two bishops and king.
3. Demonstrate understanding of opposition and outflanking through success with king vs. king exercise, and two king and pawn
exercises selected by the coach.
4. Complete “Checklist of Checkmates: Corridors” and “Checklist of Checkmates: Diagonals.”
5. Complete “Bishop Award: checkmates and tactics” worksheet. 
6. Demonstrate ability to write chess notatation

Rook: the Rook Award recognizes that the recipient has developed his or her endgame and checkmate skills, and has become a tournament player.
1. Previously earned Bishop.
2. Demonstrate understanding of Lucena (building a bridge) and Philidor (sixth rank defense) endgame positions (rooks and pawn).
3. Complete “Checklist of Checkmates: Intersections” and “Checklist of Checkmates: Knights.”
4. Complete “Rook Award: checkmates and tactics” worksheet.
5. Complete two scholastic tournaments (no voluntary byes or forfeits).

Queen: the Queen Award recognizes that the recipient has developed the habit of chess study, and has proven his or her abilities through success in tournament competition.
1. Previously earned Rook.
2. Demonstrate understanding of queen vs. pawn endgames (winning and drawing ideas).
3. Complete “Checklist of Checkmates: Combinations,” “Checklist of Checkmates: Queens,” and “Checklist of Checkmates:
Challenges.”
4. Complete “Queen Award: checkmates and tactics” worksheet.
5. Complete three scholastic tournaments (no voluntary byes or forfeits), scoring three points or more in at least one event.

King: the King Award recognizes that the recipient has become a strong scholastic player.
1. Previously earned Queen.
2. Demonstrate correct play from five opposition exercises selected by the coach.
3. Correctly solve fifteen problems in fifteen minutes selected at random from “Checklist of Checkmates” exercises with 86%
accuracy (13 of 15).
4. Show evidence of independent study of tactics exercises book, such as those by Fred Reinfeld, Bruce Pandolfini, or others (at least 100 problems solved).
4. Earn NWSRS or USCF rating above 1200.