I do Alekhine against e4, Budapest against d4-c4. If you don't like Budapest, you can try the Grunfield (I dropped it for Budapest). Against English, I just hunker down with the hedgehog formation so nothing there XD
Against Bird's opening, I play From's Gambit, simply because its a gambit
@Gramps4
I would highly recommend the Owen Defense. It's sharp, you get both bishops mobile (which is a problem in the French Defense), and it can be effectively played against both 1. e4 and 1. d4.
Three warnings -
First, don't play b6 immediately, only AFTER playing e6 first. With any luck, they'll think you're looking to play the French Defense (if 1. e4) or a Semi-slav (if 1. d4), and be perplexed after 2. ..b6.
Second, the Owen is VERY SHARP, you are fighting for the center right from the beginning. The idea is to fix and destroy white's pawn at d4, and to force white to either open the long diagonal for your light square bishop, trade light square bishops (which is usually good for black), or impede his own kingside defense (usually with a move like f3 to support the otherwise-weak e4 pawn).
Third, there is no good book study on its various lines, so you'll have to work them out yourself. What little book there is, namely Staunton's suggestion of an early c7-c5 is wrong. The pawn thrust you are preparing for is d7-d5, and you need the c-pawn back to kick out a white knight which might want to occupy b5.
Takes a while to figure out all the lines, but it is solid against everyone I play, who are mostly in the 1800-2000 range rating-wise.
Owen's Defense is not surprising. It is simply weak, and the strategy you recommend simply results in a bad French for Black. White should answer ...d5 with e5! You run a big risk of being blasted on the Kingside while doing nothing on the Queenside (the difference between the French and Owens).
The early b6-systems only work when White has played c4 (English Defense, Queen's Indian, etc) or e3 (Colle, London, etc). Lines with e4 and d4 with no c4 are a problem for Black. d4 is weaker when c4 has been played.