I thought of the term "banana split" in the late 1970's (a couple/few years after I learned backgammon). However, then I heard a couple of other players use it before it could likely have reached them through word of mouth originating from me. I think it's just one of those puns that suggests itself and we will probably never know who (or where) came up with it first.
As to what "banana split" means... I meant it to mean any play that breaks one's own inner board point in order to hit a blot, and in the process leaving two new inside blots. It doesn't necessarily mean also breaking an anchor, though I've heard it used to mean that specifically.
I suggest something like "banana split with nuts" (or with fudge or with whipped cream or with a cherry on top?) to refer to a specific type of banana split that not only goes bananas in the inner board but also breaks another point on the board, thereby creating two new non-inside blots (in addition to the two new inside blots).
Below is a rollout of the position I posted. It is somewhat similar to one that Laila Leonhardt recently posted on FB. [Blue has his 9pt, blot on 8pt, and five-point board missing the 2pt. White has three on each of her midpoint, 8pt, 6pt and 4pt, two on 1pt, blot on 23pt.] In that alternate position, the best play is a regular banana split, no nuts (well, one nut).
There is further discussion underneath the rollout.

              
 
              
 | | White is Player 2
score: 0 pip: 105 | | 7 point match | pip: 111 score: 0
Blue is Player 1 | |
| XGID=-BBaBBB--AB----c--cd--bBb-:1:-1:1:32:0:0:0:7:10 |
| Original Position, with score 0-0 to 7 |
| 1. | Rollout1 | 10/8 6/3* | eq: -0.252 |
| Player: Opponent: | 49.35% (G:14.53% B:0.08%) 50.65% (G:21.80% B:2.57%) | Conf.: ± 0.004 (-0.256...-0.247) - [99.9%] Duration: 25 minutes 49 seconds |
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| 2. | Rollout1 | 6/3* 5/3 | eq: -0.262 (-0.010) |
| Player: Opponent: | 48.15% (G:13.00% B:0.07%) 51.85% (G:17.13% B:1.39%) | Conf.: ± 0.004 (-0.266...-0.257) - [0.1%] Duration: 21 minutes 23 seconds |
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| 3. | Rollout1 | 6/4 6/3* | eq: -0.262 (-0.010) |
| Player: Opponent: | 47.74% (G:9.96% B:0.09%) 52.26% (G:12.11% B:0.85%) | Conf.: ± 0.004 (-0.266...-0.259) - [0.0%] Duration: 23 minutes 42 seconds |
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| 4. | Rollout2 | 10/7 5/3* | eq: -0.283 (-0.032) |
| Player: Opponent: | 48.31% (G:14.13% B:0.07%) 51.69% (G:22.98% B:2.65%) | Conf.: ± 0.009 (-0.292...-0.275) - [0.0%] Duration: 4 minutes 55 seconds |
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| 5. | Rollout2 | 9/6 5/3* | eq: -0.287 (-0.036) |
| Player: Opponent: | 46.74% (G:11.66% B:0.08%) 53.26% (G:12.58% B:0.90%) | Conf.: ± 0.008 (-0.296...-0.279) - [0.0%] Duration: 4 minutes 33 seconds |
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| 6. | Rollout2 | 6/3* 3/1 | eq: -0.307 (-0.055) |
| Player: Opponent: | 46.32% (G:8.94% B:0.08%) 53.68% (G:13.61% B:0.94%) | Conf.: ± 0.007 (-0.314...-0.300) - [0.0%] Duration: 4 minutes 24 seconds |
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| 7. | Rollout2 | 9/7 6/3* | eq: -0.323 (-0.071) |
| Player: Opponent: | 45.82% (G:11.75% B:0.08%) 54.18% (G:16.84% B:1.17%) | Conf.: ± 0.008 (-0.331...-0.315) - [0.0%] Duration: 3 minutes 55 seconds |
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| 8. | Rollout2 | 23/21 6/3* | eq: -0.325 (-0.073) |
| Player: Opponent: | 47.57% (G:15.51% B:0.09%) 52.43% (G:26.96% B:3.48%) | Conf.: ± 0.009 (-0.334...-0.316) - [0.0%] Duration: 4 minutes 35 seconds |
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| 9. | Rollout2 | 23/20 5/3* | eq: -0.361 (-0.109) |
| Player: Opponent: | 46.61% (G:14.74% B:0.09%) 53.39% (G:27.09% B:3.74%) | Conf.: ± 0.008 (-0.369...-0.353) - [0.0%] Duration: 4 minutes 08 seconds |
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| 10. | Rollout2 | 10/7 9/7 | eq: -0.373 (-0.121) |
| Player: Opponent: | 43.75% (G:7.09% B:0.07%) 56.25% (G:8.41% B:0.54%) | Conf.: ± 0.006 (-0.380...-0.367) - [0.0%] Duration: 3 minutes 54 seconds |
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| 11. | Rollout2 | 9/4 | eq: -0.412 (-0.160) |
| Player: Opponent: | 41.95% (G:6.77% B:0.09%) 58.05% (G:8.09% B:0.38%) | Conf.: ± 0.007 (-0.419...-0.405) - [0.0%] Duration: 3 minutes 44 seconds |
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1 5184 Games rolled with Variance Reduction. Dice Seed: 49849457 Moves: 3-ply, cube decisions: XG Roller
2 1296 Games rolled with Variance Reduction. Dice Seed: 49849457 Moves: 3-ply, cube decisions: XG Roller
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If the rollout is right, then according to my definitions, six of the top nine plays (#1, #4, #5, #7, #8 and #9) are all banana splits, and #3 meets the spirit of the term (one less point and two more blots inside). Four of the nine (#1, #4, #8 and #9) are banana splits with nuts.
For reference, below is a rollout of the same position at DMP: Again, most of the top 9 are banana splits, and this time the nutty banana splits are clustered closer to the top: #1, #2, #5 and #7.
Nack

              
 
              
 | | White is Player 2
score: 0 pip: 105 | | 1 point match | pip: 111 score: 0
Blue is Player 1 | |
| XGID=-BBaBBB--AB----c--cd--bBb-:0:0:1:32:0:0:0:1:10 |
DMP: #1, #2, #5, #6, #7 and #9 are banana splits; #1, #2, #5 and #7 are nutty |
| 1. | Rollout1 | 10/8 6/3* | eq: -0.001 |
| Player: Opponent: | 49.94% (G:9.96% B:0.02%) 50.06% (G:22.31% B:4.58%) | Conf.: ± 0.004 (-0.005...+0.003) - [100.0%] Duration: 4 minutes 55 seconds |
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| 2. | Rollout1 | 10/7 5/3* | eq: -0.019 (-0.018) |
| Player: Opponent: | 49.03% (G:10.64% B:0.04%) 50.97% (G:24.42% B:5.73%) | Conf.: ± 0.004 (-0.024...-0.015) - [0.0%] Duration: 5 minutes 13 seconds |
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| 3. | Rollout1 | 6/3* 5/3 | eq: -0.027 (-0.025) |
| Player: Opponent: | 48.67% (G:8.81% B:0.02%) 51.33% (G:17.94% B:3.08%) | Conf.: ± 0.004 (-0.031...-0.023) - [0.0%] Duration: 4 minutes 18 seconds |
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| 4. | Rollout1 | 6/4 6/3* | eq: -0.046 (-0.045) |
| Player: Opponent: | 47.68% (G:8.19% B:0.04%) 52.32% (G:12.26% B:1.07%) | Conf.: ± 0.004 (-0.050...-0.043) - [0.0%] Duration: 4 minutes 27 seconds |
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| 5. | Rollout1 | 23/21 6/3* | eq: -0.051 (-0.049) |
| Player: Opponent: | 47.47% (G:9.97% B:0.05%) 52.53% (G:26.12% B:1.77%) | Conf.: ± 0.004 (-0.054...-0.047) - [0.0%] Duration: 4 minutes 21 seconds |
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| 6. | Rollout1 | 9/6 5/3* | eq: -0.051 (-0.050) |
| Player: Opponent: | 47.45% (G:8.84% B:0.04%) 52.55% (G:13.95% B:1.28%) | Conf.: ± 0.004 (-0.055...-0.047) - [0.0%] Duration: 4 minutes 35 seconds |
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| 7. | Rollout2 | 23/20 5/3* | eq: -0.068 (-0.066) |
| Player: Opponent: | 46.62% (G:11.23% B:0.03%) 53.38% (G:28.36% B:2.73%) | Conf.: ± 0.005 (-0.072...-0.063) - [0.0%] Duration: 3 minutes 08 seconds |
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| 8. | Rollout3 | 6/3* 3/1 | eq: -0.072 (-0.071) |
| Player: Opponent: | 46.39% (G:7.91% B:0.03%) 53.61% (G:13.03% B:1.10%) | Conf.: ± 0.005 (-0.077...-0.068) - [0.0%] Duration: 3 minutes 22 seconds |
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| 9. | Rollout2 | 9/7 6/3* | eq: -0.075 (-0.073) |
| Player: Opponent: | 46.27% (G:8.30% B:0.01%) 53.73% (G:17.16% B:2.31%) | Conf.: ± 0.006 (-0.080...-0.069) - [0.0%] Duration: 2 minutes 57 seconds |
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| 10. | Rollout3 | 10/7 9/7 | eq: -0.130 (-0.128) |
| Player: Opponent: | 43.52% (G:6.01% B:0.02%) 56.48% (G:9.05% B:0.69%) | Conf.: ± 0.004 (-0.134...-0.125) - [0.0%] Duration: 2 minutes 44 seconds |
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| 11. | Rollout4 | 5/2 5/3* | eq: -0.141 (-0.140) |
| Player: Opponent: | 42.95% (G:6.07% B:0.00%) 57.05% (G:14.77% B:1.16%) | Conf.: ± 0.005 (-0.146...-0.136) - [0.0%] Duration: 3 minutes 25 seconds |
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| 12. | Rollout2 | 9/4 | eq: -0.163 (-0.162) |
| Player: Opponent: | 41.84% (G:4.40% B:0.00%) 58.16% (G:8.43% B:0.29%) | Conf.: ± 0.004 (-0.167...-0.159) - [0.0%] Duration: 2 minutes 15 seconds |
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1 1296 Games rolled with Variance Reduction. Dice Seed: 49841853 Moves: 3-ply, cube decisions: XG Roller
2 764 Games rolled with Variance Reduction. Dice Seed: 49841853 Moves: 3-ply, cube decisions: XG Roller
3 765 Games rolled with Variance Reduction. Dice Seed: 49841853 Moves: 3-ply, cube decisions: XG Roller
4 763 Games rolled with Variance Reduction. Dice Seed: 49841853 Moves: 3-ply, cube decisions: XG Roller
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