As James K said, this doesn't have anything to do with the spelling.
Different accents of English have different patterns for the use of "dark l" vs. "light l". As you mentioned, some accents are even described as having only one of these sounds. Accents with "both" sounds are often described as actually having a "gradient" of "darker" vs. "lighter" /l/s in different environments. So this is a pretty complicated question.
In general, there seems to be a tendency for "dark l" sounds to be used more often in American accents than in British accents.
In the British "Received Pronunciation" accent of English that is often taught to learners, "light l" is used before a vowel sound, even when the /l/ comes at the end of a word or after a stressed vowel. This accent would have "light l" in "killer", "bellicose", and "pullup". (I don't know if the "light l" in "pullup" might be a bit darker than the light l used in other contexts; that sounds somewhat plausible to me.)
In certain American English accents, "dark l" is used before consonants, and also word-finally, even when a vowel follows, but "light l" is used before a vowel in most word-medial contexts. These accents would have "light l" in "bellicose", but because "pullup" is a compound, I think it might have dark l. I'm also not sure how words like ""killer" are treated in these accents.
In other American English accents (like mine), it seems that "dark l" is used whenever the /l/ is not at the start of a metrical "foot" (either as the very first sound, as in "lucky", or as an element of the onset cluster in the first syllable of the foot, as in "climbing"). So I have "dark l" in killer, bellicose and pullup because the /l/ is not at the start of a foot, but I have "light l" in words like illiterate, illuminate, crystallography, collect where the /l/ is at the start of a stressed syllable.
I quoted some sources in my answer to a related ELU question: L in the middle of a word: dark l or light l?
/ε/ exists, and is dominant. If you scroll down in this chart of phoneme variations among English dialects to the row for ε, you will see that most dialects, including most British dialects, have /ε/. However, "conservative RP", Cockney, and Australian and New Zealand variants are listed as having /e/ or /e̞/.
The main Wikipedia page for the English language is simpler and simply says that the "bed" vowel is /ε/ in AmE and /e̞/ in RP.
OALD makes its own choices of pronunciation symbols, which are based on, but sometimes differ from, IPA. They actually document their system and their choices in detail, and even specifically mention the case of "bed":
Yet there are reasons to vary the phonemic
symbolization of RP. Many of the commonly used vowel symbols are frustratingly far from their
IPA values—the /e/ of being a good example. Given its close relationship with the
spelling letter the choice of symbol—especially for ELT—makes sense, but the actual IPA
value of [e] is quite different.
("ELT" means "English Language Teaching").
And, as mentioned in the comments, the vowel chart in your question is quite misleading. Compare this one instead:
in which /ε/ and /e/ are neighbors (and /e̞/, which is not on the chart, would sit in between them.
Best Answer
The speaker in your two examples is British, with the characteristic "EE" sound of that accent. The way he pronounces both "feel" and "feeling" is greatly exaggerated, in order to allow you to hear each part clearly. You should not try to imitate this way in natural speech.
I can't much help you with that accent, but I can say that a "standard" American accent will certainly pronounce the final "L", although not as if it was a second syllable. FEE-ul would be too much. End the word with your tongue pressed to the roof of your mouth, and that should be sufficient.
With "feel", You pronounce the "EE" sound fine. Your challenge is to learn to pronounce the "L" sound as distinct from the "R" sound. Once you get that, the word will sound more natural. Your second recording is better, so just learn to make that same sound more quickly, with less exaggeration.
Additionally, many English speakers will blend the final "L" of a word like "feel" with any following word that starts with "L":
And so on.
"Feeling" is two syllables, each of which is pronounced. The "L" sound between the syllables can be very brief, with only a quick touch of the tongue to the roof of the mouth.