If my boat is sinking should I bale or bail the water out?

The verb that means scooping water out of a boat is usually spelled "bail". However, the alternative spelling "bale" is also used by some writers. The "bale" spelling may be more common in the UK (the OED says bail is "Now often less correctly" spelled bale, who's etymology is an "erroneous spelling of bail"). The second spelling is not listed in all dictionaries (e.g. Merriam-Webster only has bail), so always using "bail" may be a safer pick, especially in the American English.

While either spelling is likely to be understood given a nautical context, a specific person is not likely to mix them together. If you use the base spelling "bail", then you should use "bailing" as the gerund form. If you prefer "bale", you'd use "baling". You should not write "bail" and "baling" together.

There are other verbs spelled "bail" (get somebody out of jail) and "bale" (form into a bale or bundle). If you're searching for uses of the first verb (referring to boats), you may find a lot of uses of these other verbs, so you should be careful in interpreting your results. Your search for "Baling pump" for instance, seems to turn up a lot of references to hydraulic pumps involved in "baling" machines which are used to create bales of things like straw or cardboard, not to pump water overboard on a boat.

Bail is the typical form, but bale is listed on Lexico as a variant.

The origin is described there as:

Early 17th century from obsolete bail ‘bucket’, from French baille, based on Latin bajulus ‘carrier’.

Further supported by Etymonline

"to dip water out of," 1610s, from baile (n.) "small wooden bucket" (mid-14c.), from nautical Old French baille "bucket, pail," from Medieval Latin *baiula (aquae), literally "porter of water," from Latin baiulare "to bear a burden" (see bail (n.1)).